Mastering Desmos for SAT Success

What's up everybody? Good to see you. We are going to spend a little bit of time together. I'm going to show you 20 questions that you absolutely must know how to do using Desmos. They are of varying difficulty, so a few might seem a little easy for you. Some will seem kind of on the intermediate side, and I guarantee you there will be a few where you are really going to learn a lot. They'll be very hard. Those would be the questions which you otherwise would never have been able to do without Desmos, or if you attempted to do it without Desmos. It would take a really long time. Let's see guys, three quick things and then we're going to jump right into this first question. First, please don't be passive when you watch the video. Please make sure that you have Desmos live and ready to go and follow along with me, okay? That will develop good muscle memory so that when it comes to the test coming up, by the way, on March 9 and any future edition of the digital SAT that you might take, you know exactly how to approach the question. You're not going to hesitate and you can just crush it using Desmos. Second, please get a mouse. Okay, I strongly believe that it will enable you to zoom in and out real fast as you can see that I'm doing right here, as well as pan around because it'll help you get certain features of your graph much faster. And just make sure that it has a wheel and that allows you to move around extremely quickly. Okay. And the final thing is just understand that Desmos is super powerful. We think of it for the graphing capabilities, but it also does basic calculations. So it's fine to bring your physical calculator. Most students use the TI-84 plus CE. That's totally fine, but I would not use it for basic calculations. Use Desmos. So if you had to do 54 times 5, bam, there's your answer. Why would you waste time going to the calculator and then back to your computer, right? So... The only situation where you should use your physical calculator is if you have some handy formulas stored in it, okay? Also remember that the SAT does provide a little reference sheet, and I just clicked on it, and as you can see, you get some fairly handy geometry formulas. Okay, we're ready to get started here, guys. I'm super excited. Let's attack number eight here. This is, by the way, from a Blue Book emulator that I use for my students. These questions are exactly like what you're going to see on the SAT. So we have x equals 70 over x plus 3, and we need to get the negative solution. Always remember to focus on little details in the question guide so that you don't make accidental mistakes, because there may be more than one solution. We need to make sure it's negative. So all we're going to do is just literally type the equation directly into Desmos. It will solve equations for you automatically. So x is 70 over... x plus 3. And if it seems like I'm typing a little slow and I'm having to look down and hunt for things, it's just because my microphone is directly in front of me. So do excuse me if it seems like I'm kind of like bumbling about there. So x is 70 over x plus 3. Pretty cool. Now we have two vertical bars, two vertical lines. Those do not represent the graph of the equation. They represent the solutions. And you get them by just going directly to where they hit the x-axis. And this is the one that we want. We would ignore the other. because it's positive obviously. And here we go. And we just zoom in. And you just zoom in enough where you're convinced that you know the answer. It will be negative 10 in this case, right guys? So negative 10 it is. And what do we do? We just type in negative 10. And there you go. And then just hit next. And that will commit the answer for you. Okay, guys, here is our next victim. Number 10. And this question says, given the function y equals f of x graphed in the xy plane. What is the y intercept? Okay, so again, focus on that. We want the y intercept, not the x, and I think the easiest thing to do here would be to simply copy that function directly into Desmos. Incidentally, you will not be able to on the actual test, sadly, you will not be able to copy anything from the problem and just, you know, paste it directly into Desmos. So we will just rewrite this. f of x equals 27. times two-thirds, and then we're going to raise it to the x power. Let me just get rid of that parentheses. Now, to raise something to a power, guys, you need to just go shift six, okay? Shift six, and then bam, there's your exponent. Now, what do we do next? we want to see where this graph hits the y-axis. So we just pan up a little bit. And Desmos is smart. So if you see kind of like a little gray dot and then it turns black, that essentially means that it knows where the intercept is. And there's your answer, 0,27. And we're going to pick C. Make sure that your answer lights up blue. Then hit next and that commits your answer. All right, guys, next problem, number 12. Incidentally, these questions right now that I'm doing are on module 1, so those do tend to be maybe on the slightly easier side. It says a triangle has a height of x centimeters and a base length of x minus 3 centimeters. If the triangle has an area of 104 square centimeters, what is the value of x? So you will get scrap paper on the test and it'll be completely fine. In fact, I would encourage it to maybe just sort of write the information down and set up an equation. My point, however, is once you've set up that equation, Do not do any more work by hand. Feed it into Desmos, okay? So the way it would work is as follows. We know that the formula for an area of a triangle is area equals one half base times height. Again, you could just confirm that on your reference sheet. You see it right over here, okay? So what we're going to do is we're just going to say the area is 104. I don't need to solve for the variable, okay? Desmos will do that for you. So the area is equal to, and now I'm going Set up the formula. So it's one half. So one half. The way I got that fraction, by the way, is you just use the little fraction bar. It should be directly next to the shift key on your keyboard. Now, next, what is the base? It has a base length of x minus three. So I'm just going to write x minus three in parentheses. And it has a height of x. And so I'm just going to include the x like that. And guess what, guys? We're done. Can you believe it? We're done. You might think, yeah, but where is the graph? Just hit the home button. because then that'll get you closer to where the action might be. And now we're just going to zoom out a little bit and bam, look at that. Again, two vertical lines that represents the two solutions. Obviously, we want to pick the positive one since it's a length and lengths are always positive. We're going to zoom in a little bit. And I think you guys would agree that the answer is going to be 16. This is multiple choice, though. That means you have a lot more forgiveness and you don't have to worry about getting exactly precise on locating your solution. Even though it's pretty obvious it's going to be 16 comma 0. So we're going to click B, 16, and we're going to move on. All right, guys, get ready. This one, I think, is like the first point in the test where you're going to be like, oh, everything's been going great. And like, oh, no, even in module one, they're throwing a fairly hard question at me. I think you guys are going to be blown away how easy this is. Desmos essentially hands you the answer. Let's just read it first. It says, Amelia made a deposit in a savings account at a bank and then made no additional deposits or withdrawals. The table shows the exponential relationship between the time Y in years, since Amelia's deposit, and the total amount D in dollars in the account. Which equation best represents the relationship between Y and D? Okay, now what you're going to do, okay, is you're going to notice that they told you that the relationship is exponential. We're going to create a table. that basically just copies those values directly into the table. And then guess what, guys? We're going to basically get the equation of that exponential relationship. And I'll show you exactly how we do it. So let's just get rid of this. And now to get a table, the fastest way to do it, I'm going to teach you a few shortcuts as we go. Just simply type the word table, okay? Next, we're going to enter the data. You're going to have 0, 1, and 2 for the x values. And then we're going to enter the y values, which were uh 740 um 74370 and 74742 and again forgive me i know i'm looking all over the place it's just because i've got this big microphone right in my face so we've got 740 74370 74742 perfect okay and now what are we going to do next guess what we are going to use desmos's awesome little equation um I would call it like the equation creator. Okay. So all you have to do is notice that the table was set up as X sub one and Y sub one. So you do need to type those exactly as is. So we're going to go Y sub one to get a subscript. You go shift and then the minus sign, which is, you know, on the upper layer of your keyboard. Notice it went down into subscript type of one. And then to get out of that subscript, use the right arrow. Okay. Now this is extremely important. This symbol is basically what hands you the equation that it's going to give you. You have to go shift and then the little button directly next to the number one, directly to the left of the number one. That looks like a little tilde symbol that you might see in Spanish class on top of the ends. And now we're ready to nail this thing. So we're saying that we want to get an equation where y sub 1 is going to be expressed as an exponential equation. The general form is simply a times b to the x. So that you should know. And so you would go a and then b. And now we want to raise that to the power of, so that's shift six. And then I'm going to type x sub one, sub means subscript. So remember that's shift and then the minus sign. And now look at that little one that I get there. And I'm going to scroll down a little bit. And guess what? Do you see what it gave me under parameters? It gave me the values for a and b, which means that my equation is y equals a, which is 740 times 1.005 to the x power. And that's it. You wouldn't really even need to graph it at this point. That wouldn't necessarily achieve anything. But maybe I will do that just for the fun of it. just so that you guys see that it did create a beautiful exponential equation. So 740 times 1.005, and we're going to raise that to the x power, okay? And just zoom out a little bit, and there it is, okay? So this is essentially what we got, and if you look in, if we zoom in a little bit, these are the three points, and basically what Desmos did is it took the three points from the table, and it... form-fitted the exact exponential equation to go through those three points. How cool is that? And so I think it should be pretty obvious that the answer is going to be, what is the answer going to be? It's going to be C, right? Because that's the one that says 740 times 1.005, the same as 1 plus.005 to the x power. Notice that in this particular case, I'm not worrying about the variables that they gave. Just use x1 and y1, okay? And There you go. So let's now move on to the next problem. Okay, guys, we are now on number 14. We have y equals 7x and y equals x plus 18. At how many points do the graphs intersect in the xy plane? So notice it's saying at how many. So we basically just need to count how many times these two equations may intersect one another. I'm going to give you a quick little shortcut. When you've got multiple things going on and you're ready to start a new problem, you don't have to keep hitting the X button. Just hit the little gear icon, hit delete all, and bam, you're at the top. Okay, so we're going to enter Y equals 7X, and the next equation is Y equals X plus 18. And we're going to just see where these guys intersect. And because they're two lines, it should be pretty obvious, right, that they can only intersect at that one point. Okay, we don't really care what the value is. But just so you're in case you're curious, you would click on that, that would immediately get you the value of their intersection point, which is at 3,21. But the point is, they only intersect once. And so we pick B, one, and we're done with that question. Okay, guys, moving on to our next problem. So now we are going to get the actual intersection point. So this question says 9x plus 6y is 30, 4x plus 3y is 18. And we have to get the value of x again. I can't emphasize enough, guys. You know, you are able to crush these questions in a matter of seconds. So don't waste that opportunity by picking the value of y, for example. Okay, so make it super clear what your goal of the question is and make sure you pick the corresponding answer. So we want to make sure we pick the x value, not the y value. Okay, let's get rid of these two guys. Let's type these two equations in. 9x. plus 6y, again, bear with me, silly microphone right in my face, and then we have 4x plus 3y equals 18. Okay, not 19, but 18. Okay, and notice that there's this cute little gray dot that appears, and then when it turns black, that pretty much guarantees that that's the correct intersection point. Yeah, that gray dot right there, and I just turn it black, and then if I zoom in, You can be rest assured that that's the intersection point. Okay. So what's the answer? We want x. It's negative 6. Notice that, guys, that they did put 14 in there for those students who are just a little bit too fast and are not trying to read carefully what the question is asking for. So of course, we're going to pick a negative 6. That will be the answer. And we're going to move on to the next problem. All right, guys, we're getting close to the end of the module. So you should anticipate the questions are going to gradually get a little bit harder. And let's see what we can do with this one. It says the function f is defined by f of x is 12x squared minus 30x minus 162. For what value of x does it reach its minimum value? This is an automatic. Don't even think about it. Just immediately enter that equation, that function directly into Desmos. And you're simply going to click on the vertex because the vertex always corresponds to either the minimum or the maximum value. So here we go. This is an absolute dead giveaway type question. You have to make sure you absolutely get it correct. Okay, minus 30x minus 162. And we are going to hit the home button because it might just allow us to see what's going on a little bit better. And we just zoom down. See, I can do this so fast with the mouse. And Here we go. We want to get in a little close just so we're certain that we got the correct value. And notice that gray dot turns black. There's the vertex. So the coordinates are 1.25. That would be the x value. And the y coordinate is negative 180.75. Once again, guys, let's not drop the ball. It says for what value of x does it reach its minimum? So what am I going to put? The 1.25, right? We're going to enter 1.25. And we're going to collect all the points that we deserve for that question. Okay guys, so now we are getting close to the final boss. We are at number 21. Remember, each of your math modules will have exactly 22 questions. So here we go. In the xy plane, line m passes to the point and is parallel to the line represented by the equation y equals 7x plus 5. If line m also passes to the point, what's the value of m? So there's a lot going on here. It might be kind of hard to kind of like figure out exactly how to approach it. I think the cool thing about Desmos is that you can visualize what's going on, and then you might be able to come up with a really clever idea to kind of like fast forward through the problem. And I'll show you exactly what I mean here. So the first thing we can do is just without even thinking, you have to enter that equation. So why go 7x plus 5? Because we have to make sure that the other one is going to be parallel to it. So here's 7x plus 5. Okay, perfect. Now, the next thing I want to do is line M is going to pass through the point 0, 0. So guess what? Desmos, of course, allows you to enter points. So we're going to enter the point 0, 0. However, what I'd like to do is include the other point directly next to it. And the cool thing is I'm going to connect them with a line. It'll be much easier to visualize another line parallel to that line rather than to just use two points. So you can do more than one point on the same line in your Desmos entry bar. and we're going to go comma and now we're going to enter the other point four comma n and of course the n will pop up as a slider and we definitely want to use that slider so we'll click on n and now we've got a cool little slider um but you would agree right it's not terribly useful right now because it's going to be kind of hard to figure out how that point would be parallel to the uh to the line okay so what am i going to do click on the little icon in blue with the little dots, hold it down. Okay, don't just click and then release. Hold it down. And then do you see the option at the bottom with lines? Just move that to the right so it's toggled on. And look at that, guys. You now have a line that connects the two dots, which is going to make our task much easier to figure out how this thing is going to be parallel. Now notice that as I increase my value of n, right, it is starting to get slightly more parallel to the red line, but not yet. So let's actually adjust n to maybe go from 10, because we already know that 10 didn't work, so let's now use that as a lower boundary. Notice that you have total control over the range of your values for your slider as well as the step size in case you want a little bit more precision. So I'm going to make it go from 10 to say 30, okay? We can still keep the step size as 1, that's fine. And now let's see what happens as I move this. Ah, look at that! It's getting closer and closer. And it looks to me like we're starting to laser in on this. We might want to zoom out because that might help tell me if it's parallel or not. And notice that we are starting to get pretty darn close, aren't we? Now, I will admit myself that it's a little hard to tell the difference right now between, say, 27 and 28, right? And unfortunately, because this is a student response question, we have to be exact, okay? If we put in 27... and the answer was 28, it would be wrong. With multiple choice, you have much more forgiveness, and you probably would already get the closest answer. So we have to be careful here, and because it's only two points, we don't get to see the full extent of the line. So basically what you could do is say to yourself, wait a second, I know that the slope of my red guy is what? Because it's in y equals mx plus b form. You got it, the slope is seven. So now could I just simply set up a little expression that says that the slope, of the line connecting the two dots would be, as you guys know, the difference in the y values divided by the difference in the x values. The difference in the y values would be n minus 0. So we could just write n. And then the difference in the x values would be 4 minus 0. So therefore, I'm just going to do n divided by 4. And what's my goal? My goal is to make sure that that value is 7. Because remember, any two lines that are parallel must have the same slope. So since that got me a value of 7 and that matches the 7 in the 7x plus 5, I do believe we have our answer. And the answer is 28. Okay? So, again, it's possible that some students may be able to kind of like figure that all out without Desmos. But I just sort of feel like being able to visualize it and using the tools to get that ever so close, then suddenly you realize, ah, if I really want to nail this, I can just make use of the fact that it has to have a slope of 7. And seeing that there. on the display is really going to help. So the answer will be 28 and we are going to hit next. Okay, guys, we're on 22. Guess what? This is the final boss of module one, and then we'll move on to module two. What does it say? It says, in the xy plane, a line with equation 4y equals p for some constant p intersects a parabola at exactly one point. If the parabola has equation y equals 5x squared minus 7x, what is the value of p? Okay. Again, when you've got multiple things here, just hit the gear icon, hit delete all, and now we're ready to start from scratch. First thing I would do, is type in the 4y equals p because we want to make sure that we move that line through the slider p until it hits that parabola in exactly one point. Now if we do 4y equals p, you're going to notice a problem. It didn't give us a slider and it has an error message. We only support implicit equations of x and y. So guess what? There's a trick. You can give it an x, but since x is not in the equation, just write plus zero x, okay? And now you're good to go. You can add the slider p and bam there it is. And notice that I move this up and down and that makes sense because we essentially have a horizontal line and yeah it's just moving up and down. Next thing we want to do is enter the equation of the parabola which was y equals 5x squared minus 7x. Okay there it is. Now what's our goal? Our goal is to make sure that we move that slider. so that it intersects the parabola in exactly one point. So I do need to zoom in a bit. Again, this will be much faster when you use a mouse with a wheel. And we're kind of close, right? Right now it goes through two places, but it's very close to hitting it in one spot. So I'm going to continue to move my slider. Ooh, now it's looking really good. I moved it down to negative one point, sorry, negative 9.8. And again, you must do a hard zoom to ensure that that's correct. And I think we're good. What do you guys think? There's no way. that that goes through more than one point at this point. So what are we going to enter as our answer? The value of p, which through our slider, we know is negative 9.8. Boom, done. Guess what? We're going to absolutely crush that first module. And now, guys, join me as we go into module two. OK, guys, here we are on module two. And believe it or not, I thought we would start off with number one. I know that this looks like a pretty easy question. But I do want to let you guys know that I'm going to start off with number one. Desmos has some pretty nice statistics features. So we are asked to find the median of this set. So why not just enter this data in and let Desmos do the work for you? Okay. Because, you know, they could be sneaky. And even though the data in that set is going up from left to right, they could have been sneaky and put in some numbers where they're not going from an ascending order. But the good news is Desmos will immediately sort the data for you, and it will give you the correct median no matter what. The data does not need to be. ranked from low to high. How do we get the median? Very easy. Guess what? You type the word median, okay? And then median is a function, so therefore it does require parentheses, and then we just enter the data. 21, 23, 24. Separate your data with commas, by the way. 25, 26, 28, 32, 34, and 37. Bam, you're done. Whenever you just have a single expression and it calculates it for you, it'll be in the bottom. So the answer is 26. And there you go. So we're going to hit 26. And we are going to nail that question. Real quick, guys. What if they had asked you for the mean of the data? Guess what? Desmos will also do the mean for you. Just change it instead of median. Type mean and bam. The average of that data set will be 27.7 repeating. Super handy, right? Okay. So we're going to pick B and let's move on to the next problem. Okay guys, now we are on number six. What do we have? We have 28 divided by 4x plus 32. Then they say the given expression is equivalent to 7 over x plus r, where r is a constant and x is greater than zero. What is the value of r? Ah, what's going on here? Well, I'll tell you what's going on here. Another opportunity to use Desmos. Anytime they have a question where they say, which of the following expressions is equivalent to the one that they're given, and you don't really see the algebra that you need to do, or there's some weird trick, don't worry. You can almost always use Desmos to help you solve it. So how are we going to attack this? We are going to attack it by basically taking the 28 over 4x plus 32, and we're just going to write it as a function. So we can just say y equals 28 over 4x plus 32. Okay, let's hit the home so that we can get better viewing of our graph. And there it is. There's your graph. Incidentally, guys, just a little thing. If you don't like your colors because the black sort of merges with the rest of the screen, you can change your colors easily. Just hold down and then pick your favorite color. Okay. I like blue. And so we can always change the colors if you'd like to. So there is the equation. of, or the graph rather, of 28 over 4x plus 32. How do you know that you can get the right answer? You will get the right answer if it matches directly over that graph. In other words, if it's a direct overlay. You'll see exactly what I mean. So let's try choice A and see if R, if it were equal to 8, would be equivalent to the existing graph in blue. So what am I going to do? I'm just going to do y equals 7. Over x plus, again, I'm just using the strategy of testing the answers, right? Nobody says we have to be a superhero and figure out what r is on our own. So I'm going to enter the 8. And ooh, look at that, guys. Did you notice something? The blue graph disappeared. What does that mean? It means that since the red graph supplanted it, it's guaranteed to be the answer. Because if I just even were to click the little icon, do you notice how it goes on and off? It's like blinking. Whenever that happens, it means that they are identical. It means that that is the equivalent expression. It's the one when r equals 8. You do not need to test any other answer choices. That would be a complete waste. The answer is a8. Click the next button, and let's move on. Okay, guys, here is another question. I think this is a super easy question, but it's different from what we've done. And I just want you to know that Desmos can evaluate any function that you give it. By evaluate, what I mean is you can put anything you want into it. Numbers are a piece of cake. So let's just enter our equation g of x equals 6 plus x cubed. Again, to get the exponent, do shift 6. There's my cube, and there's my function. And now to figure out what g of 3 is, you just are simply going to type g of 3. And again, because it's just an expression and it can evaluate it, your answer appears in the bottom right corner. The answer is 33. You're going to pick 33, and you're going to move on. All right, guys, guess what? Word problems? Did you think that, uh... We wouldn't attack a couple word problems with Desmos? You bet. Of course we can do it. Here we go. Let's read this one. It says the function h of t equals negative 16t squared plus 96t plus 100 represents an object launched from a platform where h of t is the height of the object above the ground in feet t seconds after it is launched. Which number corresponds to the height in feet from which the object was launched? Okay, I'm going to graph this. I'm going to show you another cool little trick that could be handy in certain situations. Okay, so first thing we do is we're going to enter this function. As a rule of thumb, you want to typically use x as your main variable. You can use any letter you want for the name of your function. H is fine. However, I didn't have any problem when I typed in t, so as long as you get a graph, you're fine. You can use that variable. So I think we can use h of t equals, and then let's type in this equation, negative 16t squared plus 96t. plus 100. Okay and here's the thing. There's a lot going on here and there's some stuff to the left of the y-axis which implies that there's some negative numbers but since this is launching an object and time would be starting at zero and larger we can actually add a restriction and the way you do that is just use curly braces like this and just say t greater than or equal to zero. To do greater than or equal you go greater and then the equal sign, and it immediately makes a greater than or equal sign. Of course, you can use the little keys on the virtual keyboard on Desmos, but I find the more you can do by just typing it, the faster. Okay? So we want it to be greater than zero, and do you notice how it chopped off the left part of the graph? Isn't that cool? It just allows us to focus on the more relevant information. Now, it wants the corresponding height from which the object was launched. I think you'd agree, guys, that... When you launch the object, that happened exactly at time zero. So therefore, it's basically just saying, what is the y-intercept? And there it is. It's 0, 100. And so we are going to pick 100. And that is the end of the problem. Incidentally, if they had asked for at what time does it reach its maximum height, then, of course, you would click on the maximum of the graph. And it would be at, let's see here, 3,244. So in other words... Three seconds into the flight, it reached a height of 244. Okay, pretty useful information. Let's go ahead and move on to the next problem. Okay, guys, check this one out. This is a super cool one. You will be blown away how easy Desmos makes this. So here we go. Excuse me. We have the function g given by x squared minus four. What we're going to do is we're going to type g of x equals x squared minus four. Okay, and then guess what? Anytime you do a function, of course you can see it, right? There it is in all its glory. But you can also create a table. This would be the perfect opportunity to create a table because all the answer choices are giving you tables. Those tables are showing you what values of x correspond to the y values or the g of x values for this particular function. So rather than wasting time trying to plug stuff in, just let Desmos do the work for you. So hit the little gear icon and then just click on table. Okay, so create table. Bam! Look at this nice little table. And we can adjust the values for x so that they match the answer choices. We don't really care about the negative 2, the negative 1, do we? No, we don't. So we're just going to type 0, and notice that it'll immediately update the corresponding value of g, which is negative 4, and then we're going to put in the 2. Again, it's updated to give you a 0, and then we're going to put in the 4, and bam, we have our g of x values are going to be negative 4, 0, and 12. And I think you guys would agree that the only answer choice that shows that would be negative 4, 0, 12. Look at that. Super fast. These questions, guys, once you start to get the hang of it, are probably only going to take you about 20 to 30 seconds. Okay, guys, check it out. We are going to do number 12. This is in a module one again. And what do we have? We have a cone has a volume of exactly two pi cubic inches. What is the height of the cone in inches? So what I'm going to demonstrate here is how even a geometry problem, you can actually feed it directly into Desmos as long as there's some governing equation. In this case, of course there is. There's got to be a formula for the volume of a cone. Just check your reference sheet, because remember, there's a lot of geometry formulas, and you see it right there. Volume is equal to one-third pi r squared h. Perfect. So now, watch what we can do. We can let Desmos solve this for us so that we don't get entangled in any numbers or algebra. It'll just give it to us automatically. So here we go. The volume is 2 pi. The shortcut for pi, by the way, is pi. So just type pi. And that's the volume. But that, of course, would be equal to one third. So we go one third. And then it's pi again, pi. And then it's r squared h. Now, you can tell from the diagram that the base is two. The cone is flipped, but it has a base of two. But the radius of that circular base would be one, right? So we just do one third times pi. And, you know, just to make it clear, in the formula. I'm going to go one squared. I know that one squared is one, but we'll still put that there anyway, just so you see the formula. And what's left is the height. Now, very important, don't use h because if you attempted to, it's not going to give you anything. Use x. x is the magic variable so that you can get those pretty vertical lines to represent your solution. And bam, there it is. Okay. So it looks like if we zoom in, it's going to be six, right? Yes, indeed. Every now and then it seems like Desmos will occasionally give you a nice dot that you can be convinced is correct on the x-axis. And other times it doesn't. It's a little bit whimsical with that. But in any case, we're sure that the answer is 6. So you're going to type in 6. And that is the height of the cone. All right, guys. A brand new problem here that we can easily let Desmos solve for us. It says 3 times 6x plus 11 is 9 times 9x plus 4. In the given equation, d is a constant. if the equation has no solution. So make sure that you, you know, even on your scrap paper, just write that down so that you don't forget. You want it to make sure that there has no solution. The question is, what is the value of d? So in this case, you definitely want to look at this from a more graphical viewpoint. So what you could do is say to yourself, wait a second, I could set the left side of that equation, say equal to y, it would graph a line. And then I could take the right side of that equation, also say set it equal to y. Then that'll give you two lines. How do you know if two lines have no solutions? I think we've done this before. You have to make sure that they are parallel to one another. Okay? So let's go ahead and type in y equals 3 times 6x plus 11. All right? And there it is. Okay? And then we're going to type in the other one. y equals, now I'm going to do d. It'll give me a slider. Okay? Anything other than x and y that you enter as an equation. or rather as a function, will pop up as a slider. So that's perfect. That's exactly what we want. 9x plus 4. Now we're going to take our slider d, and pretty cool, right? We're already kind of close to having no solutions, but you see that they do intersect here, don't they? So let's see what happens if I move my d away. Perfect. And it looks like 2 almost did the job. And what I would do is just keep scrolling a little bit just to double check. And I think you guys would agree, look, I'm already at negative 100. They haven't gotten closer to one another one bit. Those guys are dead parallel. So you can be rest assured, nothing to worry about. They are parallel. So what does that mean? The answer is going to be 2 because that's the magic value for D that ensures that those two lines have no equation, which means that the, sorry, that they have no solution, which means that the original equation has no solution when D is 0. All right, a brand new problem. Did you think that Desmos wouldn't be able to... Do inequalities? Now, of course it can do inequalities. So what are we going to do? We're going to try to find out which point, which is one of our answer choices, is a solution to the given system of inequalities in the xy plane. Let's type in the first one. So we've got y less than 2x plus 6. And if I just hit the home button, there it is. So what that means, guys, is that the white region, the white space, any point there would not be a solution. Anything in that dark brown region, is a point that would be a solution to that first equation. But we have another one that makes it a system. And so now we go y less than negative 1 third x plus 8. OK, pretty cool, right? So now your second equation is in red. And anything shaded red, which would be below that dashed line, would be a point or a solution. But we want it to solve the system. So a system is the combination of those two inequalities. And where do you think that would be? If you said, yes, it's going to be in that dark, dark red region, you're 100% correct, because that represents where there's overlap between the two solutions. Now, how do we figure out which one of these answer choices is correct? It couldn't be easier. You're just going to simply test A first. And we know how to do points. You just do parentheses, 8 comma 4, end parentheses. Look at that beautiful blue dot. It's dead. smack in that dark red region, which means it would be a valid solution for both inequalities. So you pick A, and then we move on. Okay, guys, let's see what's next. We've got a few more to go, and we will be done with this video. I was hoping to keep it under an hour, and I think we're on pace for that. And great job to all of you who have stuck with me through all these problems. Hopefully you did it with Desmos, just like I did. Now when you see these problems pop up on your first SAT, I think you guys are going to know exactly how to attack them. Here we go. A brand new type of problem. Again, Desmos will handle statistics. We've got 3558912. It says if an integer x is added to the data set above, the mean and median of the data set will be equal. What is a possible value of x? Okay, so let's go for it. Do you remember that we already had a list of data and we were able to easily get the median of it? So we're going to do the same thing. However, I want to... teach you guys an easier way to deal with this data. I think what would be smart is to put it in a list. And the way you do a list is you just use brackets and you then enter your numbers and you just separate them with a comma. So go 3, 5, 5, 8, 9, 12. And there it is. And it's pretty cool. It confirms that it's a six element list. Just remember lists must be between brackets. Now, what you should do is, it's not mandatory, but I would recommend naming it a variable. Then that way, you don't have to recopy the list all the time. You can just feed the variable to the mean and the median. Let's call it A. So A is equal to that list of data, okay? If we were to calculate the mean of that, we would just type mean, and we will need this. So let's type mean of A. Again, you need the parentheses after mean, because mean is a function. And then we're going to do median. also of a. Now, as it stands, the mean is 7 and the median is 6.5. However, we need to add a value x to this list such that when we do that, the mean and the median will be equal. So what's our strategy? We want to add another number, and we should do it as a variable. Now, if you do x, don't do it. Okay? X is best used for graphing purposes. And we're not really thinking about graphs right now. We want it just to do it purely in terms of the numbers. And it's just a complete mess. I wouldn't even know how to interpret all that. So instead, just give it a non-graphing variable. The graphing variables, of course, are x and y. Just give it b, and that'll act as a slider. And then you just click on b. And of course, you could use any other letter, c, d, and so on. So we're going to use b. And notice that... we have an opportunity to slide it, don't we? Okay. Now we do have to be precise here because this is not a multiple choice. If it were a multiple choice, we could just feed those values directly in for b and test them. Here we have to be exact. Now I'm trying to move my b value such that the mean and the median are the same. Right now when b is 3, for example, you get 6.4 for the mean and 5 for the median. That won't do it, will it? So I'm going to move that slider. It's chopping it off a little bit so that I don't, you can't quite see the screen with both values, but we're getting close at 7.1 and 7.8. Let's see here. Let's just check what happened when it was eight. Okay. So what I'm going to do guys, just because it's kind of blocking the view of both of them, let's just expand this and you'll always be able to expand your Desmos. on the actual test. Now we can see everything going on here. Now it's going to be super easy. So 6.8, we're very close right now, and bam, there it is. Okay, when b is 7, notice that the mean is 7 and the median is 7, and therefore the answer is 7. We'll put the little calculator back the way it was, and so you will type 7 into your student response, and you should be 100% confident that you know you got that correct. Okay guys, another word problem, and here we go. We have an online newspaper and it has 1,642 subscribers when it launches. The newspaper gains popularity such that the number of subscribers triples every six months. Okay, so just remember that it multiplies by three every six months. The formula S equals 1,642 times three to the NT power, and that gives the number of subscribers T years after launch. So we have to be a little bit careful with the units. It triples every six months, but the T in the formula is in years. The question is, what is the value of N? This is an absolute piece of cake using Desmos. Let's first eliminate everything we've done so far. And as a rule of thumb, as soon as you see like a formula or something, just put that in immediately. Now, again, I would probably, I would probably, if you wanted to graph it, then you should probably do it as like F of X. So that's fine. Let's just go F of X. Okay, so we're just changing the S to F of X. That's not a big deal. 1642. This is very important, guys. Do not use commas and numbers. Notice that the formula in the question says 1642 with a comma. Don't put that in Desmos. Otherwise, it'll sort of almost treat it as if it were a point. So I just type 1642, 3, and then we're going to raise it to the power of. I want N as my slider, and I'm just going to use X since I declared it as F of X. But X, as you can see, is representing my time. We're going to add our slider. Okay. And there's the graph, which is, you know, kind of cool. And now how do I know what the correct answer is? Well, what I'm going to do is I'm going to write F of zero. And that means that the initial number of newspapers was 1642. And now here's the cool part. I'm going to evaluate the function at F of 0.5. Now, why on earth would I do that? Because it triples every six months. Six months is half a year, and since the t is in years, I need to calculate f of 0.5, okay? That would be six months or half a year later. How do I know if it's the correct answer? If the f of 0.5 value is three times bigger than the f of 0. All right, so now what I want to do is just go to my slider, and I'm going to change the 1 to 1 sixth. Let's see what happens when n is 1 sixth, okay? F of 0 is 1642, and F of 0.5 is 1799. I think you guys can tell that's definitely not three times bigger, okay? So it's not A. By the way, one thing that you should get familiar with is the crossout tool. Do you see this little ABC icon up here? If you click that, you have the option of getting rid of answers, and I think that's actually kind of helpful. In this case, it's not very helpful at all because it just makes the 1 6 look like another fraction, but usually it crosses it out perfectly. All right, so what am I going to do next? Of course, I'm just going to try choice B. So now we're going to see if one half works. And if I change it to a half, what happens to my f of 0? Of course, that doesn't change. But my f of 0.5 becomes 2160. Is 2160 three times bigger than 1642? Of course, you could check with the calculator, but it's not. OK, so therefore, we can cross out B. And now let's try C. I'm going to change my n value now to just simply 2. And what happens now? Our F of 0 is still 1642. And ooh, this looks promising, right guys? F of 0.5 is 4926, okay? Incidentally, so let's just pretend we wanted to check to see if that's equal to 3, if that ratio is equal to 3. So you don't have to retype numbers. You can actually just click on the cell that has that number, Control-C, and then Control-V directly into the next line. And bam, there's your 4926. We want to divide that by. The previous value, which was 1642, same idea. Click it. It comes up kind of blue. Hit Control-C. And then directly in the denominator of this fraction, hit Control-V. And bam! Look at that, guys. It's equal to 3. So we do know indeed that the answer is going to be C2. How about that? That only required about five lines of work in Desmos. It couldn't be easier. That problem would be very, very difficult to try to do any other way by hand. Let's move on to the next problem. Okay, guys, so we have for number 22, when graphed in the xy plane, line g passes to the points negative 6, 3, and 2, negative 1. Line h is the image of line g. By the way, image is just a fancy way of saying that the line got moved somewhere. Line h is the image of line g shifted up five units in the xy plane. What is the x-intercept of line h? Let me ask you a question, guys. Would you agree? If we knew exactly what the equation for line G was, our task would be a lot easier. Because imagine you had that equation, and it was like an mx plus b form. Then if it got shifted up five units, we would just simply add five to that, and then check where it hits the x-axis. Well, guess what? We can definitely get the equation of that line. Do you remember how we easily got the equation when we had three points, and we had to get the exponential graph? We can do the exact same thing for lines. How do we do it? Very easy. Type table. and enter your data. So in the x column, you want your negative 6. In the y, it will be 3. And then move down to the next row. It will be 2 and negative 1. And those two dots up here, which is kind of cool. And now we want to get the equation of the line that goes to those two dots. What are you going to type below here? You're going to go y underscore 1. That gets you the subscript. Remember, it's shift and then the minus sign. And then use the arrow to get out of the subscript. Type shift tilde. So shift the button next to the number one. And then just type m for slope, x underscore one, and then plus b. Okay. And now check it out. If you scroll down, do you see it says parameters? Guess what? Those are the missing values for the line. So in other words, the line is essentially equal to negative 0.5x plus zero. Remember? The slope intercept form of a line is y equals mx plus b. Okay, so I could enter y equals negative 0.5x plus b, which we know is 0. So I know it's a little silly for me to write it, but that's okay. And now remember, line h is a brand new line, and it's the image of the previous one shifted up 5 units. So what am I going to do? Of course, just add 5. And there it is. It's the red line that just got shifted up. Once again, I can't emphasize enough, guys. Don't just quickly, blindly click that.05 because that would not be the correct answer. That's the y-intercept. So just slide over and move over to the x-intercept. And bam, it's 10, 0. So just enter your answer of 10. That's the x-intercept. And guess what, guys? We have one more question to go. So please join me for the final boss. Okay, guys, here we go. This video is going to end up probably being... exactly an hour. I hope you guys got a lot out of it. We are going to do this final question, and then we're going to call it a day. And of course, I'm going to wish you guys all a lot of luck on your first SAT. Check this out. This is a classic word problem. This would drive most kids absolutely nuts. By the way, this is number 22 on module two. So it's just supposed to be the hardest problem. I think you're going to agree that Desmos is going to make a joke out of this problem. Here we go. A beverage dispenser contains 80 ounces of fruit juice. The juice was made by mixing x ounces of grape juice with y ounces of apple juice. Okay? The value of y is 8 more than twice the value of x. How many ounces of apple juice are in the punch? We're going to nail this. It's basically just a classic system of equations. We're going to type those equations directly into Desmos. We're going to see where the intersection point is, and that will get us the apple juice. Okay? Notice that apple juice is going to be our y value. grape juice is going to be the x value. So what's the first equation that we could write? Well, we have 80 ounces of fruit juice, right? And we know that we're using x ounces of grape juice and y ounces of apple juice. So of course, we could write the equation x plus y has to add up to a total of 80, okay? And there is my first equation, okay, drawn there in red. Let's get the second equation. That's from the third sentence. It says the value of y, so that would just be y. is, that's equals, 8 more than, so that would be plus 8, more than twice the value of x. So wouldn't that just be 2x? plus 8, okay? And now we just zoom in and we grab the intersection point. And the intersection point is at 2456, which makes perfect sense because those two numbers do indeed add up to 80. And it gives you the correct proportions to make this problem work. And the question is, how many ounces of apple juice? Again, I can't emphasize enough, guys. Don't drop the ball at the last minute after doing all this amazing work using Desmos to crush these questions for you. Make sure you enter the correct number. Which one is it? Is it 24 or 56? Well, since the apple juice was given to us as the Y value, then of course we need to enter 56. Okay. So 56 would be the number of ounces of apple juice in this punch. And of course, if they had asked for the grape juice, it would have been 24. So there you go, guys. I hope you enjoyed. That takes care of this little video to show you a bunch of questions that you are very likely to see or something similar. And I do believe that now you have some really powerful tools that you can use on Desmos to have an amazing test day. With that in mind, I wish you all the best and have a great test. See you guys.

What's up everybody? Good to see you. We are going to spend a little bit of time together.

I'm going to show you 20 questions that you absolutely must know how to do using Desmos. They are of varying difficulty, so a few might seem a little easy for you. Some will seem kind of on the intermediate side, and I guarantee you there will be a few where you are really going to learn a lot. They'll be very hard. Those would be the questions which you otherwise would never have been able to do without Desmos, or if you attempted to do it without Desmos.

It would take a really long time. Let's see guys, three quick things and then we're going to jump right into this first question. First, please don't be passive when you watch the video. Please make sure that you have Desmos live and ready to go and follow along with me, okay?

That will develop good muscle memory so that when it comes to the test coming up, by the way, on March 9 and any future edition of the digital SAT that you might take, you know exactly how to approach the question. You're not going to hesitate and you can just crush it using Desmos. Second, please get a mouse. Okay, I strongly believe that it will enable you to zoom in and out real fast as you can see that I'm doing right here, as well as pan around because it'll help you get certain features of your graph much faster.

And just make sure that it has a wheel and that allows you to move around extremely quickly. Okay. And the final thing is just understand that Desmos is super powerful.

We think of it for the graphing capabilities, but it also does basic calculations. So it's fine to bring your physical calculator. Most students use the TI-84 plus CE. That's totally fine, but I would not use it for basic calculations.

Use Desmos. So if you had to do 54 times 5, bam, there's your answer. Why would you waste time going to the calculator and then back to your computer, right? So... The only situation where you should use your physical calculator is if you have some handy formulas stored in it, okay?

Also remember that the SAT does provide a little reference sheet, and I just clicked on it, and as you can see, you get some fairly handy geometry formulas. Okay, we're ready to get started here, guys. I'm super excited.

Let's attack number eight here. This is, by the way, from a Blue Book emulator that I use for my students. These questions are exactly like what you're going to see on the SAT.

So we have x equals 70 over x plus 3, and we need to get the negative solution. Always remember to focus on little details in the question guide so that you don't make accidental mistakes, because there may be more than one solution. We need to make sure it's negative.

So all we're going to do is just literally type the equation directly into Desmos. It will solve equations for you automatically. So x is 70 over...

x plus 3. And if it seems like I'm typing a little slow and I'm having to look down and hunt for things, it's just because my microphone is directly in front of me. So do excuse me if it seems like I'm kind of like bumbling about there. So x is 70 over x plus 3. Pretty cool. Now we have two vertical bars, two vertical lines. Those do not represent the graph of the equation.

They represent the solutions. And you get them by just going directly to where they hit the x-axis. And this is the one that we want. We would ignore the other.

because it's positive obviously. And here we go. And we just zoom in.

And you just zoom in enough where you're convinced that you know the answer. It will be negative 10 in this case, right guys? So negative 10 it is.

And what do we do? We just type in negative 10. And there you go. And then just hit next.

And that will commit the answer for you. Okay, guys, here is our next victim. Number 10. And this question says, given the function y equals f of x graphed in the xy plane. What is the y intercept?

Okay, so again, focus on that. We want the y intercept, not the x, and I think the easiest thing to do here would be to simply copy that function directly into Desmos. Incidentally, you will not be able to on the actual test, sadly, you will not be able to copy anything from the problem and just, you know, paste it directly into Desmos. So we will just rewrite this.

f of x equals 27. times two-thirds, and then we're going to raise it to the x power. Let me just get rid of that parentheses. Now, to raise something to a power, guys, you need to just go shift six, okay?

Shift six, and then bam, there's your exponent. Now, what do we do next? we want to see where this graph hits the y-axis.

So we just pan up a little bit. And Desmos is smart. So if you see kind of like a little gray dot and then it turns black, that essentially means that it knows where the intercept is. And there's your answer, 0,27. And we're going to pick C.

Make sure that your answer lights up blue. Then hit next and that commits your answer. All right, guys, next problem, number 12. Incidentally, these questions right now that I'm doing are on module 1, so those do tend to be maybe on the slightly easier side.

It says a triangle has a height of x centimeters and a base length of x minus 3 centimeters. If the triangle has an area of 104 square centimeters, what is the value of x? So you will get scrap paper on the test and it'll be completely fine. In fact, I would encourage it to maybe just sort of write the information down and set up an equation. My point, however, is once you've set up that equation, Do not do any more work by hand.

Feed it into Desmos, okay? So the way it would work is as follows. We know that the formula for an area of a triangle is area equals one half base times height.

Again, you could just confirm that on your reference sheet. You see it right over here, okay? So what we're going to do is we're just going to say the area is 104. I don't need to solve for the variable, okay? Desmos will do that for you.

So the area is equal to, and now I'm going Set up the formula. So it's one half. So one half. The way I got that fraction, by the way, is you just use the little fraction bar. It should be directly next to the shift key on your keyboard.

Now, next, what is the base? It has a base length of x minus three. So I'm just going to write x minus three in parentheses. And it has a height of x.

And so I'm just going to include the x like that. And guess what, guys? We're done.

Can you believe it? We're done. You might think, yeah, but where is the graph? Just hit the home button. because then that'll get you closer to where the action might be.

And now we're just going to zoom out a little bit and bam, look at that. Again, two vertical lines that represents the two solutions. Obviously, we want to pick the positive one since it's a length and lengths are always positive.

We're going to zoom in a little bit. And I think you guys would agree that the answer is going to be 16. This is multiple choice, though. That means you have a lot more forgiveness and you don't have to worry about getting exactly precise on locating your solution.

Even though it's pretty obvious it's going to be 16 comma 0. So we're going to click B, 16, and we're going to move on. All right, guys, get ready. This one, I think, is like the first point in the test where you're going to be like, oh, everything's been going great.

And like, oh, no, even in module one, they're throwing a fairly hard question at me. I think you guys are going to be blown away how easy this is. Desmos essentially hands you the answer.

Let's just read it first. It says, Amelia made a deposit in a savings account at a bank and then made no additional deposits or withdrawals. The table shows the exponential relationship between the time Y in years, since Amelia's deposit, and the total amount D in dollars in the account.

Which equation best represents the relationship between Y and D? Okay, now what you're going to do, okay, is you're going to notice that they told you that the relationship is exponential. We're going to create a table.

that basically just copies those values directly into the table. And then guess what, guys? We're going to basically get the equation of that exponential relationship. And I'll show you exactly how we do it. So let's just get rid of this.

And now to get a table, the fastest way to do it, I'm going to teach you a few shortcuts as we go. Just simply type the word table, okay? Next, we're going to enter the data. You're going to have 0, 1, and 2 for the x values.

And then we're going to enter the y values, which were uh 740 um 74370 and 74742 and again forgive me i know i'm looking all over the place it's just because i've got this big microphone right in my face so we've got 740 74370 74742 perfect okay and now what are we going to do next guess what we are going to use desmos's awesome little equation um I would call it like the equation creator. Okay. So all you have to do is notice that the table was set up as X sub one and Y sub one.

So you do need to type those exactly as is. So we're going to go Y sub one to get a subscript. You go shift and then the minus sign, which is, you know, on the upper layer of your keyboard. Notice it went down into subscript type of one.

And then to get out of that subscript, use the right arrow. Okay. Now this is extremely important.

This symbol is basically what hands you the equation that it's going to give you. You have to go shift and then the little button directly next to the number one, directly to the left of the number one. That looks like a little tilde symbol that you might see in Spanish class on top of the ends. And now we're ready to nail this thing.

So we're saying that we want to get an equation where y sub 1 is going to be expressed as an exponential equation. The general form is simply a times b to the x. So that you should know.

And so you would go a and then b. And now we want to raise that to the power of, so that's shift six. And then I'm going to type x sub one, sub means subscript.

So remember that's shift and then the minus sign. And now look at that little one that I get there. And I'm going to scroll down a little bit.

And guess what? Do you see what it gave me under parameters? It gave me the values for a and b, which means that my equation is y equals a, which is 740 times 1.005 to the x power. And that's it. You wouldn't really even need to graph it at this point.

That wouldn't necessarily achieve anything. But maybe I will do that just for the fun of it. just so that you guys see that it did create a beautiful exponential equation. So 740 times 1.005, and we're going to raise that to the x power, okay?

And just zoom out a little bit, and there it is, okay? So this is essentially what we got, and if you look in, if we zoom in a little bit, these are the three points, and basically what Desmos did is it took the three points from the table, and it... form-fitted the exact exponential equation to go through those three points. How cool is that?

And so I think it should be pretty obvious that the answer is going to be, what is the answer going to be? It's going to be C, right? Because that's the one that says 740 times 1.005, the same as 1 plus.005 to the x power.

Notice that in this particular case, I'm not worrying about the variables that they gave. Just use x1 and y1, okay? And There you go.

So let's now move on to the next problem. Okay, guys, we are now on number 14. We have y equals 7x and y equals x plus 18. At how many points do the graphs intersect in the xy plane? So notice it's saying at how many. So we basically just need to count how many times these two equations may intersect one another.

I'm going to give you a quick little shortcut. When you've got multiple things going on and you're ready to start a new problem, you don't have to keep hitting the X button. Just hit the little gear icon, hit delete all, and bam, you're at the top.

Okay, so we're going to enter Y equals 7X, and the next equation is Y equals X plus 18. And we're going to just see where these guys intersect. And because they're two lines, it should be pretty obvious, right, that they can only intersect at that one point. Okay, we don't really care what the value is.

But just so you're in case you're curious, you would click on that, that would immediately get you the value of their intersection point, which is at 3,21. But the point is, they only intersect once. And so we pick B, one, and we're done with that question. Okay, guys, moving on to our next problem.

So now we are going to get the actual intersection point. So this question says 9x plus 6y is 30, 4x plus 3y is 18. And we have to get the value of x again. I can't emphasize enough, guys.

You know, you are able to crush these questions in a matter of seconds. So don't waste that opportunity by picking the value of y, for example. Okay, so make it super clear what your goal of the question is and make sure you pick the corresponding answer. So we want to make sure we pick the x value, not the y value.

Okay, let's get rid of these two guys. Let's type these two equations in. 9x.

plus 6y, again, bear with me, silly microphone right in my face, and then we have 4x plus 3y equals 18. Okay, not 19, but 18. Okay, and notice that there's this cute little gray dot that appears, and then when it turns black, that pretty much guarantees that that's the correct intersection point. Yeah, that gray dot right there, and I just turn it black, and then if I zoom in, You can be rest assured that that's the intersection point. Okay.

So what's the answer? We want x. It's negative 6. Notice that, guys, that they did put 14 in there for those students who are just a little bit too fast and are not trying to read carefully what the question is asking for. So of course, we're going to pick a negative 6. That will be the answer.

And we're going to move on to the next problem. All right, guys, we're getting close to the end of the module. So you should anticipate the questions are going to gradually get a little bit harder.

And let's see what we can do with this one. It says the function f is defined by f of x is 12x squared minus 30x minus 162. For what value of x does it reach its minimum value? This is an automatic.

Don't even think about it. Just immediately enter that equation, that function directly into Desmos. And you're simply going to click on the vertex because the vertex always corresponds to either the minimum or the maximum value.

So here we go. This is an absolute dead giveaway type question. You have to make sure you absolutely get it correct. Okay, minus 30x minus 162. And we are going to hit the home button because it might just allow us to see what's going on a little bit better.

And we just zoom down. See, I can do this so fast with the mouse. And Here we go. We want to get in a little close just so we're certain that we got the correct value. And notice that gray dot turns black.

There's the vertex. So the coordinates are 1.25. That would be the x value. And the y coordinate is negative 180.75. Once again, guys, let's not drop the ball.

It says for what value of x does it reach its minimum? So what am I going to put? The 1.25, right?

We're going to enter 1.25. And we're going to collect all the points that we deserve for that question. Okay guys, so now we are getting close to the final boss.

We are at number 21. Remember, each of your math modules will have exactly 22 questions. So here we go. In the xy plane, line m passes to the point and is parallel to the line represented by the equation y equals 7x plus 5. If line m also passes to the point, what's the value of m? So there's a lot going on here.

It might be kind of hard to kind of like figure out exactly how to approach it. I think the cool thing about Desmos is that you can visualize what's going on, and then you might be able to come up with a really clever idea to kind of like fast forward through the problem. And I'll show you exactly what I mean here.

So the first thing we can do is just without even thinking, you have to enter that equation. So why go 7x plus 5? Because we have to make sure that the other one is going to be parallel to it.

So here's 7x plus 5. Okay, perfect. Now, the next thing I want to do is line M is going to pass through the point 0, 0. So guess what? Desmos, of course, allows you to enter points. So we're going to enter the point 0, 0. However, what I'd like to do is include the other point directly next to it. And the cool thing is I'm going to connect them with a line.

It'll be much easier to visualize another line parallel to that line rather than to just use two points. So you can do more than one point on the same line in your Desmos entry bar. and we're going to go comma and now we're going to enter the other point four comma n and of course the n will pop up as a slider and we definitely want to use that slider so we'll click on n and now we've got a cool little slider um but you would agree right it's not terribly useful right now because it's going to be kind of hard to figure out how that point would be parallel to the uh to the line okay so what am i going to do click on the little icon in blue with the little dots, hold it down.

Okay, don't just click and then release. Hold it down. And then do you see the option at the bottom with lines?

Just move that to the right so it's toggled on. And look at that, guys. You now have a line that connects the two dots, which is going to make our task much easier to figure out how this thing is going to be parallel.

Now notice that as I increase my value of n, right, it is starting to get slightly more parallel to the red line, but not yet. So let's actually adjust n to maybe go from 10, because we already know that 10 didn't work, so let's now use that as a lower boundary. Notice that you have total control over the range of your values for your slider as well as the step size in case you want a little bit more precision. So I'm going to make it go from 10 to say 30, okay?

We can still keep the step size as 1, that's fine. And now let's see what happens as I move this. Ah, look at that!

It's getting closer and closer. And it looks to me like we're starting to laser in on this. We might want to zoom out because that might help tell me if it's parallel or not. And notice that we are starting to get pretty darn close, aren't we?

Now, I will admit myself that it's a little hard to tell the difference right now between, say, 27 and 28, right? And unfortunately, because this is a student response question, we have to be exact, okay? If we put in 27...

and the answer was 28, it would be wrong. With multiple choice, you have much more forgiveness, and you probably would already get the closest answer. So we have to be careful here, and because it's only two points, we don't get to see the full extent of the line.

So basically what you could do is say to yourself, wait a second, I know that the slope of my red guy is what? Because it's in y equals mx plus b form. You got it, the slope is seven.

So now could I just simply set up a little expression that says that the slope, of the line connecting the two dots would be, as you guys know, the difference in the y values divided by the difference in the x values. The difference in the y values would be n minus 0. So we could just write n. And then the difference in the x values would be 4 minus 0. So therefore, I'm just going to do n divided by 4. And what's my goal? My goal is to make sure that that value is 7. Because remember, any two lines that are parallel must have the same slope.

So since that got me a value of 7 and that matches the 7 in the 7x plus 5, I do believe we have our answer. And the answer is 28. Okay? So, again, it's possible that some students may be able to kind of like figure that all out without Desmos. But I just sort of feel like being able to visualize it and using the tools to get that ever so close, then suddenly you realize, ah, if I really want to nail this, I can just make use of the fact that it has to have a slope of 7. And seeing that there. on the display is really going to help.

So the answer will be 28 and we are going to hit next. Okay, guys, we're on 22. Guess what? This is the final boss of module one, and then we'll move on to module two.

What does it say? It says, in the xy plane, a line with equation 4y equals p for some constant p intersects a parabola at exactly one point. If the parabola has equation y equals 5x squared minus 7x, what is the value of p?

Okay. Again, when you've got multiple things here, just hit the gear icon, hit delete all, and now we're ready to start from scratch. First thing I would do, is type in the 4y equals p because we want to make sure that we move that line through the slider p until it hits that parabola in exactly one point. Now if we do 4y equals p, you're going to notice a problem. It didn't give us a slider and it has an error message.

We only support implicit equations of x and y. So guess what? There's a trick. You can give it an x, but since x is not in the equation, just write plus zero x, okay?

And now you're good to go. You can add the slider p and bam there it is. And notice that I move this up and down and that makes sense because we essentially have a horizontal line and yeah it's just moving up and down. Next thing we want to do is enter the equation of the parabola which was y equals 5x squared minus 7x.

Okay there it is. Now what's our goal? Our goal is to make sure that we move that slider. so that it intersects the parabola in exactly one point. So I do need to zoom in a bit.

Again, this will be much faster when you use a mouse with a wheel. And we're kind of close, right? Right now it goes through two places, but it's very close to hitting it in one spot. So I'm going to continue to move my slider.

Ooh, now it's looking really good. I moved it down to negative one point, sorry, negative 9.8. And again, you must do a hard zoom to ensure that that's correct.

And I think we're good. What do you guys think? There's no way. that that goes through more than one point at this point. So what are we going to enter as our answer?

The value of p, which through our slider, we know is negative 9.8. Boom, done. Guess what? We're going to absolutely crush that first module. And now, guys, join me as we go into module two.

OK, guys, here we are on module two. And believe it or not, I thought we would start off with number one. I know that this looks like a pretty easy question. But I do want to let you guys know that I'm going to start off with number one. Desmos has some pretty nice statistics features.

So we are asked to find the median of this set. So why not just enter this data in and let Desmos do the work for you? Okay. Because, you know, they could be sneaky. And even though the data in that set is going up from left to right, they could have been sneaky and put in some numbers where they're not going from an ascending order.

But the good news is Desmos will immediately sort the data for you, and it will give you the correct median no matter what. The data does not need to be. ranked from low to high.

How do we get the median? Very easy. Guess what?

You type the word median, okay? And then median is a function, so therefore it does require parentheses, and then we just enter the data. 21, 23, 24. Separate your data with commas, by the way.

25, 26, 28, 32, 34, and 37. Bam, you're done. Whenever you just have a single expression and it calculates it for you, it'll be in the bottom. So the answer is 26. And there you go.

So we're going to hit 26. And we are going to nail that question. Real quick, guys. What if they had asked you for the mean of the data?

Guess what? Desmos will also do the mean for you. Just change it instead of median.

Type mean and bam. The average of that data set will be 27.7 repeating. Super handy, right? Okay.

So we're going to pick B and let's move on to the next problem. Okay guys, now we are on number six. What do we have?

We have 28 divided by 4x plus 32. Then they say the given expression is equivalent to 7 over x plus r, where r is a constant and x is greater than zero. What is the value of r? Ah, what's going on here? Well, I'll tell you what's going on here.

Another opportunity to use Desmos. Anytime they have a question where they say, which of the following expressions is equivalent to the one that they're given, and you don't really see the algebra that you need to do, or there's some weird trick, don't worry. You can almost always use Desmos to help you solve it. So how are we going to attack this?

We are going to attack it by basically taking the 28 over 4x plus 32, and we're just going to write it as a function. So we can just say y equals 28 over 4x plus 32. Okay, let's hit the home so that we can get better viewing of our graph. And there it is.

There's your graph. Incidentally, guys, just a little thing. If you don't like your colors because the black sort of merges with the rest of the screen, you can change your colors easily. Just hold down and then pick your favorite color. Okay.

I like blue. And so we can always change the colors if you'd like to. So there is the equation. of, or the graph rather, of 28 over 4x plus 32. How do you know that you can get the right answer?

You will get the right answer if it matches directly over that graph. In other words, if it's a direct overlay. You'll see exactly what I mean. So let's try choice A and see if R, if it were equal to 8, would be equivalent to the existing graph in blue. So what am I going to do?

I'm just going to do y equals 7. Over x plus, again, I'm just using the strategy of testing the answers, right? Nobody says we have to be a superhero and figure out what r is on our own. So I'm going to enter the 8. And ooh, look at that, guys. Did you notice something?

The blue graph disappeared. What does that mean? It means that since the red graph supplanted it, it's guaranteed to be the answer.

Because if I just even were to click the little icon, do you notice how it goes on and off? It's like blinking. Whenever that happens, it means that they are identical.

It means that that is the equivalent expression. It's the one when r equals 8. You do not need to test any other answer choices. That would be a complete waste. The answer is a8. Click the next button, and let's move on.

Okay, guys, here is another question. I think this is a super easy question, but it's different from what we've done. And I just want you to know that Desmos can evaluate any function that you give it. By evaluate, what I mean is you can put anything you want into it. Numbers are a piece of cake.

So let's just enter our equation g of x equals 6 plus x cubed. Again, to get the exponent, do shift 6. There's my cube, and there's my function. And now to figure out what g of 3 is, you just are simply going to type g of 3. And again, because it's just an expression and it can evaluate it, your answer appears in the bottom right corner.

The answer is 33. You're going to pick 33, and you're going to move on. All right, guys, guess what? Word problems?

Did you think that, uh... We wouldn't attack a couple word problems with Desmos? You bet.

Of course we can do it. Here we go. Let's read this one.

It says the function h of t equals negative 16t squared plus 96t plus 100 represents an object launched from a platform where h of t is the height of the object above the ground in feet t seconds after it is launched. Which number corresponds to the height in feet from which the object was launched? Okay, I'm going to graph this.

I'm going to show you another cool little trick that could be handy in certain situations. Okay, so first thing we do is we're going to enter this function. As a rule of thumb, you want to typically use x as your main variable. You can use any letter you want for the name of your function.

H is fine. However, I didn't have any problem when I typed in t, so as long as you get a graph, you're fine. You can use that variable. So I think we can use h of t equals, and then let's type in this equation, negative 16t squared plus 96t.

plus 100. Okay and here's the thing. There's a lot going on here and there's some stuff to the left of the y-axis which implies that there's some negative numbers but since this is launching an object and time would be starting at zero and larger we can actually add a restriction and the way you do that is just use curly braces like this and just say t greater than or equal to zero. To do greater than or equal you go greater and then the equal sign, and it immediately makes a greater than or equal sign. Of course, you can use the little keys on the virtual keyboard on Desmos, but I find the more you can do by just typing it, the faster.

Okay? So we want it to be greater than zero, and do you notice how it chopped off the left part of the graph? Isn't that cool?

It just allows us to focus on the more relevant information. Now, it wants the corresponding height from which the object was launched. I think you'd agree, guys, that...

When you launch the object, that happened exactly at time zero. So therefore, it's basically just saying, what is the y-intercept? And there it is.

It's 0, 100. And so we are going to pick 100. And that is the end of the problem. Incidentally, if they had asked for at what time does it reach its maximum height, then, of course, you would click on the maximum of the graph. And it would be at, let's see here, 3,244.

So in other words... Three seconds into the flight, it reached a height of 244. Okay, pretty useful information. Let's go ahead and move on to the next problem.

Okay, guys, check this one out. This is a super cool one. You will be blown away how easy Desmos makes this.

So here we go. Excuse me. We have the function g given by x squared minus four. What we're going to do is we're going to type g of x equals x squared minus four.

Okay, and then guess what? Anytime you do a function, of course you can see it, right? There it is in all its glory.

But you can also create a table. This would be the perfect opportunity to create a table because all the answer choices are giving you tables. Those tables are showing you what values of x correspond to the y values or the g of x values for this particular function. So rather than wasting time trying to plug stuff in, just let Desmos do the work for you. So hit the little gear icon and then just click on table.

Okay, so create table. Bam! Look at this nice little table.

And we can adjust the values for x so that they match the answer choices. We don't really care about the negative 2, the negative 1, do we? No, we don't.

So we're just going to type 0, and notice that it'll immediately update the corresponding value of g, which is negative 4, and then we're going to put in the 2. Again, it's updated to give you a 0, and then we're going to put in the 4, and bam, we have our g of x values are going to be negative 4, 0, and 12. And I think you guys would agree that the only answer choice that shows that would be negative 4, 0, 12. Look at that. Super fast. These questions, guys, once you start to get the hang of it, are probably only going to take you about 20 to 30 seconds. Okay, guys, check it out. We are going to do number 12. This is in a module one again.

And what do we have? We have a cone has a volume of exactly two pi cubic inches. What is the height of the cone in inches? So what I'm going to demonstrate here is how even a geometry problem, you can actually feed it directly into Desmos as long as there's some governing equation. In this case, of course there is.

There's got to be a formula for the volume of a cone. Just check your reference sheet, because remember, there's a lot of geometry formulas, and you see it right there. Volume is equal to one-third pi r squared h.

Perfect. So now, watch what we can do. We can let Desmos solve this for us so that we don't get entangled in any numbers or algebra.

It'll just give it to us automatically. So here we go. The volume is 2 pi.

The shortcut for pi, by the way, is pi. So just type pi. And that's the volume. But that, of course, would be equal to one third.

So we go one third. And then it's pi again, pi. And then it's r squared h.

Now, you can tell from the diagram that the base is two. The cone is flipped, but it has a base of two. But the radius of that circular base would be one, right?

So we just do one third times pi. And, you know, just to make it clear, in the formula. I'm going to go one squared.

I know that one squared is one, but we'll still put that there anyway, just so you see the formula. And what's left is the height. Now, very important, don't use h because if you attempted to, it's not going to give you anything.

Use x. x is the magic variable so that you can get those pretty vertical lines to represent your solution. And bam, there it is. Okay. So it looks like if we zoom in, it's going to be six, right?

Yes, indeed. Every now and then it seems like Desmos will occasionally give you a nice dot that you can be convinced is correct on the x-axis. And other times it doesn't.

It's a little bit whimsical with that. But in any case, we're sure that the answer is 6. So you're going to type in 6. And that is the height of the cone. All right, guys. A brand new problem here that we can easily let Desmos solve for us.

It says 3 times 6x plus 11 is 9 times 9x plus 4. In the given equation, d is a constant. if the equation has no solution. So make sure that you, you know, even on your scrap paper, just write that down so that you don't forget. You want it to make sure that there has no solution.

The question is, what is the value of d? So in this case, you definitely want to look at this from a more graphical viewpoint. So what you could do is say to yourself, wait a second, I could set the left side of that equation, say equal to y, it would graph a line.

And then I could take the right side of that equation, also say set it equal to y. Then that'll give you two lines. How do you know if two lines have no solutions? I think we've done this before. You have to make sure that they are parallel to one another.

Okay? So let's go ahead and type in y equals 3 times 6x plus 11. All right? And there it is. Okay?

And then we're going to type in the other one. y equals, now I'm going to do d. It'll give me a slider.

Okay? Anything other than x and y that you enter as an equation. or rather as a function, will pop up as a slider. So that's perfect.

That's exactly what we want. 9x plus 4. Now we're going to take our slider d, and pretty cool, right? We're already kind of close to having no solutions, but you see that they do intersect here, don't they? So let's see what happens if I move my d away.

Perfect. And it looks like 2 almost did the job. And what I would do is just keep scrolling a little bit just to double check. And I think you guys would agree, look, I'm already at negative 100. They haven't gotten closer to one another one bit.

Those guys are dead parallel. So you can be rest assured, nothing to worry about. They are parallel. So what does that mean? The answer is going to be 2 because that's the magic value for D that ensures that those two lines have no equation, which means that the, sorry, that they have no solution, which means that the original equation has no solution when D is 0. All right, a brand new problem.

Did you think that Desmos wouldn't be able to... Do inequalities? Now, of course it can do inequalities. So what are we going to do? We're going to try to find out which point, which is one of our answer choices, is a solution to the given system of inequalities in the xy plane.

Let's type in the first one. So we've got y less than 2x plus 6. And if I just hit the home button, there it is. So what that means, guys, is that the white region, the white space, any point there would not be a solution.

Anything in that dark brown region, is a point that would be a solution to that first equation. But we have another one that makes it a system. And so now we go y less than negative 1 third x plus 8. OK, pretty cool, right? So now your second equation is in red.

And anything shaded red, which would be below that dashed line, would be a point or a solution. But we want it to solve the system. So a system is the combination of those two inequalities. And where do you think that would be?

If you said, yes, it's going to be in that dark, dark red region, you're 100% correct, because that represents where there's overlap between the two solutions. Now, how do we figure out which one of these answer choices is correct? It couldn't be easier. You're just going to simply test A first. And we know how to do points.

You just do parentheses, 8 comma 4, end parentheses. Look at that beautiful blue dot. It's dead.

smack in that dark red region, which means it would be a valid solution for both inequalities. So you pick A, and then we move on. Okay, guys, let's see what's next. We've got a few more to go, and we will be done with this video.

I was hoping to keep it under an hour, and I think we're on pace for that. And great job to all of you who have stuck with me through all these problems. Hopefully you did it with Desmos, just like I did. Now when you see these problems pop up on your first SAT, I think you guys are going to know exactly how to attack them. Here we go.

A brand new type of problem. Again, Desmos will handle statistics. We've got 3558912. It says if an integer x is added to the data set above, the mean and median of the data set will be equal. What is a possible value of x? Okay, so let's go for it.

Do you remember that we already had a list of data and we were able to easily get the median of it? So we're going to do the same thing. However, I want to... teach you guys an easier way to deal with this data.

I think what would be smart is to put it in a list. And the way you do a list is you just use brackets and you then enter your numbers and you just separate them with a comma. So go 3, 5, 5, 8, 9, 12. And there it is.

And it's pretty cool. It confirms that it's a six element list. Just remember lists must be between brackets. Now, what you should do is, it's not mandatory, but I would recommend naming it a variable. Then that way, you don't have to recopy the list all the time.

You can just feed the variable to the mean and the median. Let's call it A. So A is equal to that list of data, okay?

If we were to calculate the mean of that, we would just type mean, and we will need this. So let's type mean of A. Again, you need the parentheses after mean, because mean is a function.

And then we're going to do median. also of a. Now, as it stands, the mean is 7 and the median is 6.5. However, we need to add a value x to this list such that when we do that, the mean and the median will be equal. So what's our strategy?

We want to add another number, and we should do it as a variable. Now, if you do x, don't do it. Okay? X is best used for graphing purposes. And we're not really thinking about graphs right now.

We want it just to do it purely in terms of the numbers. And it's just a complete mess. I wouldn't even know how to interpret all that. So instead, just give it a non-graphing variable.

The graphing variables, of course, are x and y. Just give it b, and that'll act as a slider. And then you just click on b.

And of course, you could use any other letter, c, d, and so on. So we're going to use b. And notice that...

we have an opportunity to slide it, don't we? Okay. Now we do have to be precise here because this is not a multiple choice. If it were a multiple choice, we could just feed those values directly in for b and test them. Here we have to be exact.

Now I'm trying to move my b value such that the mean and the median are the same. Right now when b is 3, for example, you get 6.4 for the mean and 5 for the median. That won't do it, will it? So I'm going to move that slider.

It's chopping it off a little bit so that I don't, you can't quite see the screen with both values, but we're getting close at 7.1 and 7.8. Let's see here. Let's just check what happened when it was eight.

Okay. So what I'm going to do guys, just because it's kind of blocking the view of both of them, let's just expand this and you'll always be able to expand your Desmos. on the actual test. Now we can see everything going on here.

Now it's going to be super easy. So 6.8, we're very close right now, and bam, there it is. Okay, when b is 7, notice that the mean is 7 and the median is 7, and therefore the answer is 7. We'll put the little calculator back the way it was, and so you will type 7 into your student response, and you should be 100% confident that you know you got that correct. Okay guys, another word problem, and here we go. We have an online newspaper and it has 1,642 subscribers when it launches.

The newspaper gains popularity such that the number of subscribers triples every six months. Okay, so just remember that it multiplies by three every six months. The formula S equals 1,642 times three to the NT power, and that gives the number of subscribers T years after launch. So we have to be a little bit careful with the units.

It triples every six months, but the T in the formula is in years. The question is, what is the value of N? This is an absolute piece of cake using Desmos.

Let's first eliminate everything we've done so far. And as a rule of thumb, as soon as you see like a formula or something, just put that in immediately. Now, again, I would probably, I would probably, if you wanted to graph it, then you should probably do it as like F of X.

So that's fine. Let's just go F of X. Okay, so we're just changing the S to F of X.

That's not a big deal. 1642. This is very important, guys. Do not use commas and numbers. Notice that the formula in the question says 1642 with a comma.

Don't put that in Desmos. Otherwise, it'll sort of almost treat it as if it were a point. So I just type 1642, 3, and then we're going to raise it to the power of.

I want N as my slider, and I'm just going to use X since I declared it as F of X. But X, as you can see, is representing my time. We're going to add our slider. Okay.

And there's the graph, which is, you know, kind of cool. And now how do I know what the correct answer is? Well, what I'm going to do is I'm going to write F of zero.

And that means that the initial number of newspapers was 1642. And now here's the cool part. I'm going to evaluate the function at F of 0.5. Now, why on earth would I do that?

Because it triples every six months. Six months is half a year, and since the t is in years, I need to calculate f of 0.5, okay? That would be six months or half a year later.

How do I know if it's the correct answer? If the f of 0.5 value is three times bigger than the f of 0. All right, so now what I want to do is just go to my slider, and I'm going to change the 1 to 1 sixth. Let's see what happens when n is 1 sixth, okay? F of 0 is 1642, and F of 0.5 is 1799. I think you guys can tell that's definitely not three times bigger, okay?

So it's not A. By the way, one thing that you should get familiar with is the crossout tool. Do you see this little ABC icon up here?

If you click that, you have the option of getting rid of answers, and I think that's actually kind of helpful. In this case, it's not very helpful at all because it just makes the 1 6 look like another fraction, but usually it crosses it out perfectly. All right, so what am I going to do next?

Of course, I'm just going to try choice B. So now we're going to see if one half works. And if I change it to a half, what happens to my f of 0?

Of course, that doesn't change. But my f of 0.5 becomes 2160. Is 2160 three times bigger than 1642? Of course, you could check with the calculator, but it's not.

OK, so therefore, we can cross out B. And now let's try C. I'm going to change my n value now to just simply 2. And what happens now? Our F of 0 is still 1642. And ooh, this looks promising, right guys?

F of 0.5 is 4926, okay? Incidentally, so let's just pretend we wanted to check to see if that's equal to 3, if that ratio is equal to 3. So you don't have to retype numbers. You can actually just click on the cell that has that number, Control-C, and then Control-V directly into the next line. And bam, there's your 4926. We want to divide that by.

The previous value, which was 1642, same idea. Click it. It comes up kind of blue.

Hit Control-C. And then directly in the denominator of this fraction, hit Control-V. And bam! Look at that, guys. It's equal to 3. So we do know indeed that the answer is going to be C2.

How about that? That only required about five lines of work in Desmos. It couldn't be easier.

That problem would be very, very difficult to try to do any other way by hand. Let's move on to the next problem. Okay, guys, so we have for number 22, when graphed in the xy plane, line g passes to the points negative 6, 3, and 2, negative 1. Line h is the image of line g.

By the way, image is just a fancy way of saying that the line got moved somewhere. Line h is the image of line g shifted up five units in the xy plane. What is the x-intercept of line h?

Let me ask you a question, guys. Would you agree? If we knew exactly what the equation for line G was, our task would be a lot easier.

Because imagine you had that equation, and it was like an mx plus b form. Then if it got shifted up five units, we would just simply add five to that, and then check where it hits the x-axis. Well, guess what?

We can definitely get the equation of that line. Do you remember how we easily got the equation when we had three points, and we had to get the exponential graph? We can do the exact same thing for lines.

How do we do it? Very easy. Type table. and enter your data. So in the x column, you want your negative 6. In the y, it will be 3. And then move down to the next row.

It will be 2 and negative 1. And those two dots up here, which is kind of cool. And now we want to get the equation of the line that goes to those two dots. What are you going to type below here? You're going to go y underscore 1. That gets you the subscript.

Remember, it's shift and then the minus sign. And then use the arrow to get out of the subscript. Type shift tilde.

So shift the button next to the number one. And then just type m for slope, x underscore one, and then plus b. Okay.

And now check it out. If you scroll down, do you see it says parameters? Guess what?

Those are the missing values for the line. So in other words, the line is essentially equal to negative 0.5x plus zero. Remember?

The slope intercept form of a line is y equals mx plus b. Okay, so I could enter y equals negative 0.5x plus b, which we know is 0. So I know it's a little silly for me to write it, but that's okay. And now remember, line h is a brand new line, and it's the image of the previous one shifted up 5 units.

So what am I going to do? Of course, just add 5. And there it is. It's the red line that just got shifted up. Once again, I can't emphasize enough, guys. Don't just quickly, blindly click that.05 because that would not be the correct answer.

That's the y-intercept. So just slide over and move over to the x-intercept. And bam, it's 10, 0. So just enter your answer of 10. That's the x-intercept.

And guess what, guys? We have one more question to go. So please join me for the final boss.

Okay, guys, here we go. This video is going to end up probably being... exactly an hour. I hope you guys got a lot out of it.

We are going to do this final question, and then we're going to call it a day. And of course, I'm going to wish you guys all a lot of luck on your first SAT. Check this out. This is a classic word problem. This would drive most kids absolutely nuts.

By the way, this is number 22 on module two. So it's just supposed to be the hardest problem. I think you're going to agree that Desmos is going to make a joke out of this problem.

Here we go. A beverage dispenser contains 80 ounces of fruit juice. The juice was made by mixing x ounces of grape juice with y ounces of apple juice. Okay? The value of y is 8 more than twice the value of x.

How many ounces of apple juice are in the punch? We're going to nail this. It's basically just a classic system of equations. We're going to type those equations directly into Desmos. We're going to see where the intersection point is, and that will get us the apple juice.

Okay? Notice that apple juice is going to be our y value. grape juice is going to be the x value.

So what's the first equation that we could write? Well, we have 80 ounces of fruit juice, right? And we know that we're using x ounces of grape juice and y ounces of apple juice. So of course, we could write the equation x plus y has to add up to a total of 80, okay? And there is my first equation, okay, drawn there in red.

Let's get the second equation. That's from the third sentence. It says the value of y, so that would just be y.

is, that's equals, 8 more than, so that would be plus 8, more than twice the value of x. So wouldn't that just be 2x? plus 8, okay? And now we just zoom in and we grab the intersection point. And the intersection point is at 2456, which makes perfect sense because those two numbers do indeed add up to 80. And it gives you the correct proportions to make this problem work.

And the question is, how many ounces of apple juice? Again, I can't emphasize enough, guys. Don't drop the ball at the last minute after doing all this amazing work using Desmos to crush these questions for you.

Make sure you enter the correct number. Which one is it? Is it 24 or 56?

Well, since the apple juice was given to us as the Y value, then of course we need to enter 56. Okay. So 56 would be the number of ounces of apple juice in this punch. And of course, if they had asked for the grape juice, it would have been 24. So there you go, guys.

I hope you enjoyed. That takes care of this little video to show you a bunch of questions that you are very likely to see or something similar. And I do believe that now you have some really powerful tools that you can use on Desmos to have an amazing test day. With that in mind, I wish you all the best and have a great test.

See you guys.

Transcript for:Mastering Desmos for SAT Success

Transcript for:
Mastering Desmos for SAT Success