all right so let's get into this EOC review here so question number one says what type of function is represented by each table so let's look at table number one we've got our X values here and all increasing by one we've got our f of x values our outputs and we can see that they are increasing by it looks like three every time and so from the increase of three every time we can say that this is going to be a linear function since the first difference is the same first difference is the same all right let's move on to table number two so table two we've got uh the x value is also increasing by one again and then we've got this time let's see if we can figure out some addition here this would be plus three and this would be plus 12. so don't think that's going to work um so let's try something else um we could try maybe multiplying by four so times four and then four times four is going to be 16 that'll work and then times four again yeah so it looks like we're going to be multiplying by 4 every time all right so for multiplying by four this is going to be an exponential function so you're going to want to make sure that you know how to tell the difference between a linear function an exponential function and a quadratic function those are the three main functions that'll be on your EOC all right say look at number two it says evaluate the function for f of negative 3 so when we are evaluating adding value they want us to take this value here and replace it with X or where we see an X we're going to put negative 3. all right so f of negative three here we go we've got negative 2 and whenever I substitute a value in I want to use parentheses so I've got minus 3 times another minus three plus one and we can use the calculator for this if we wanted to uh to check but if you also are able to do this without a calculator that's cool too so negative 3 squared is going to be positive 9 minus three well let's go ahead and do this if I treat this as a negative 3 times a negative three I'm going to have plus nine or I could say that this is minus a negative 9. so either way it turns into plus nine then I get plus one so last thing here let's say we got negative eighteen plus nine plus one negative eighteen plus nine is negative nine plus one that's going to be negative eight so we find out that f of negative three is negative eight for that function there all right key features what are the key features of the quadratic function graphed below so let's see they want the vertex so the vertex is the highest or the lowest point on a parabola basically where it turns so we've got a Vertex up here this would be a maximum and it looks like it's at negative one and four so negative one four so our axis of symmetry comes right from the value of H which is the x coordinate of the vertex and so we have x equals negative one domain and range so the domain of a quadratic function is going to be all real numbers the range in this case we have a maximum and our graph is pointing down so the highest value we have is up here at four everything else is below four so we'll say Y is less than or equal to 4. and it's equal to there because we are including four we have a point F4 that vertex there's going to be four all right so end behavior we have as X goes to Infinity on the right side or let's actually start with negative infinity and as X goes to positive Infinity what are the Y values doing well this is a parabola so both sides are doing the same thing we see both sides of the graph are pointing down so the Y values are going to negative infinity or we could say decreasing that are decreasing they're going to negative Infinity either one of those would be correct all right so now let's talk about increasing and decreasing intervals when we're talking about increasing I always imagine like a roller coaster right so we're we're going up on the function all the way here until we hit the vertex once we hit the vertex we're going to be decreasing the roller coaster is going down so how do we write the increasing interval well we know it's going to change at um not 4 it's going to change it negative 1. so this is the tricky part about increasing and decreasing it changes at the x value so we want to think about that so we're increasing before we get to negative one so we'll say when X is less than or equal to negative one well we can just say less than X is less than negative one and then decreasing is going to be when X is greater than negative one so that's kind of like the left side right side deal there all right moving right along so let's talk about solving the system of equations so we have this system here and if we're up to me I would use elimination because it's kind of set up for me to use elimination and when I use elimination I want to eliminate a variable so I want to take one of these equations and change something about it so that when I combine the equations one of my variables will eliminate so if I could take one of these two x's and make it negative then that would work so I'm just going to take that first equation multiplied by negative one so I'd have negative 2x minus 5y equals negative 14. so now I can take these two equations add that together the X variables eliminate to zero negative that's negative 1y minus 5y is negative 6y and that is equal to 2 minus 14 which is negative 12. and so we can solve for y there by dividing by negative six we get Y is equal to positive 2. so that's one of our answers and now we'll take that number and we will go plug it in to or substitute into one of the equations so it doesn't matter where I could plug it into either or any of those three equations that I have up there I guess I'll plug it into the second one so let's see take this one here I'm going to have 2X minus parentheses 2 equals 2. and so what do I do from here well I'm going to add 2 to both sides I get 2x is equal to 4 divided by 2. and I end up with x equals 2. all right so what is the solution to the system well the solution is going to be 2 2. all right next one up simplify the problem below using only positive exponents all right this is fun okay so remember negative exponents they kind of change direction based off of where they are so up here if we already have a negative exponent this X is going to move to the denominator and these are going to move up to the numerator and their exponents will become positive so let's just go with that real quick so again we're going to move x to the negative fourth that'll become x to the positive four and I become y to the positive five the three down here that didn't move at all and then we have x to the third power there so now we can still simplify this because we have x to the fourth in the numerator x to the third in the denominator and so we can simplify x times x times x and basically we're going to be taking away three X's from the X to the fourth and so we have x times y to the fifth over three so if you're confused about that x to the fourth and x to the third piece we got x times x times x times x in the top basically we have x times x times x in the bottom and so that goes to 1 that goes to one that goes to one and we just have one X left and so that's what we would end up with all right number six determine the value of the expression write your exact answer with one radical okay so we want exact we want one radical so first thing we should notice is that we have cube roots here so when you think about cube roots you want to think about your perfect cubes so what are my perfect cubes well I got one I've got because one times one times one is one two times two times two is eight 3 times 3 times 3 is 27 so I think we're going to use eight here and so because the reason I know that is because 8 goes into both 40 and 32. so let's think about this if I was going to simplify this I still have the 4 that doesn't change but now I could write that cube root of 40 as if I just take 40 for example that is 8 times 5. and so I still have the cube root of those things but now I made them two radicals instead of just one and then 32 well that's 8 times 4. and so I'll do the cube root of 8 times the cube root of 4. so now the reason why I'm doing that there is because I know that cube root of 8 is just 2. so that means now I can simplify this even more so I have 4 times 2. times that other two and then I have times the cube root of 5. times the E cube root of 4. so now let's see what we end up with here so 4 times 2 times 2 that's 8 times 2 which is 16. I can go ahead and multiply the 5 into 4 which is going to give me 20. so I have 16 times the cube root of 20 and I can't simplify the cube root of 20 anymore and so that's as simplified as I can get and I only have one radical so there we go moon right along we got number seven and they want us to graph the equation so let's set up our X and Y axis here put my x-axis there or my Y axis there and I'll put my x-axis here I guess all right so when I have a linear function all right this is a linear so I have a y equals MX plus b so that's slope intercept form so my slope is going to be 3 4. my y-intercept is going to be negative 2. so I'm going to start with that so I have minus 2 is where I put my y-intercept and my slope is three over four remember that's rise over run change in y or the change in X and so we can go up three from that Y intercept one two three and then we can run 4. and we're going to run to the right because this is a positive slope so one two three one two three four and that's we have a point there and if we wanted to do the opposite we could go down three and left 4. and so that would be the graph of our equation so now remember your graph is going to be on the computer so um on this EOC so make sure that uh you're getting some practice in with your computer graphing skills all right number eight so two linear functions are given below we've got f of x and we've got G of x which function has the greater slope all right so we know that the slope of f of x is negative three slope of G of X we want to know the change in y or the change in X so how is y changing well we're increasing by two and the X's are increasing by one and so our slope here is going to be two over one which is just two so which one is greater well that's going to be G of x all right which function has the greater y-intercept let's see y-intercept remember that's when X is equal to zero so we know in f of x that the Y intercept is 2. but what is the Y intercept for G of X because when we only have when X is one in the table so if we were to go back to when X is zero we'd have to decrease another two and then put us at 3. so we have a three we have a two so G of X is also the one with the greater y-intercept cool cool all right so now the last question here is which function has the greater x intercept so that's when y equals zero or in this case f of x and G of X and so let's see how we can do this here we can take zero set it equal to negative three X plus two remember this is for f of x and we can solve for x so we'll subtract two got negative two equals negative three x divide by negative three and we get X is equal to positive two-thirds so that's going to be our x intercept is positive two-thirds so now let's go back to G of X and if we try to write an equation for G of X we already found the slope is two the Y intercept is three so we could say G of X is equal to two X plus three and so now we're going to take 0 set it equal to 2x plus 3 and this is for G of x and we'll do the same thing so we've got negative 3 if we subtract 3 from both sides equals two x divide by 2. and we get X is equal to negative three over two so which one has the greater x intercept well this time that is going to be f of x all right that's cool question a lot of things they could ask you about two different functions number nine write the inequality that represents the graph below all right so we know this is inequality this is a line so it's we've got a linear inequality so Y is something and then an X plus b that's something there's going to be our inequality we just got to figure out which one that is all right so let's see here I'll put a box there so then we're going to be zero all right so what do we know about this function well we know that the Y intercept is negative two so B would equal negative two and then let's figure out the slope so we can go up to and then left two and that'll put us there so plus two minus two so the slope is going to be positive 2 over negative 2 which is actually negative one so we're going to have y is something and then negative 1X minus two or just negative x minus two so let's think about this here so because we're in slope intercept form if we write this in slope intercept form the direction of the shading is going to basically tell you your inequality sign so since we're shading above the line everything that's shaded is above that's going to be greater than and it's a dashed line it's not solid and so we're not going to have the equal bar because that would be a solid line so this is going to be greater than so our equation is going to be Y is greater than negative x minus 2 or inequality is going to be Y is greater than negative x minus 2. all right let's take a look at number 10. solve the equations we've got an absolute value equation we're going to start this off the same way we would with any equation if that was like parentheses instead of absolute value um we're going to divide everything by negative 3. I don't want to distribute that in I want to divide both sides by negative 3. that's going to save me some time so these will simplify to one negative 6 divided by negative three I have absolute value of 4X plus 2 is equal to positive 2. then I'm going to set up my absolute value so remember absolute value equations I have one equation that's equal to 2 and the other equation that's equal to negative two because if inside of the absolute value I had negative 2 well the absolute value of negative 2 is just two so I have 4X plus 2 equals 2. and then 4X Plus 2. equals negative 2. and the steps to solve are going to be the same we're going to subtract 2 from both sides and I'm just doing both equations at the same time and I get 4X equals 0 and over here I get 4X equals negative 4. so when I divide by 4 for both equations I get x equals zero on the left equation and I get x equals negative 1. on the right equation so there are my two answers foreign all right number 11. we deposit 350 into an account that earns simple interest at an annual rate of six and a half percent if no other deposits withdrawals were made that's the total amount of money in the account at the end of ten years simple interest formula is a equals capital P times 1 plus r times t okay so uh let's think about what we have here well we have P that's going to be 350. we have 1 plus our rate is six and a half percent so six point five percent as a decimal is point zero six five so that's what we're going to have point zero six five times ten and that's going to be our simple interest formula so if we simplify this here well yeah let's see that's going to be our amount so okay so we have 350 times and then we'll just do this in the calculator point zero six five times ten it's going to be 0.65 and so we have 1.65 because when we add that to one we get 1 plus 0.65 so 1.65 times 350 we will have 577.50 at the end of 10 years so there you go number 12. you and your friend go to a store where all the shirts cost the same amount and all the payments cost same amount okay so U buy four shirts five pairs of pants for 150 so let's write an equation for you if x is the number of shirts and Y is the number of pants we would have 4X plus 5y and we spent 150 dollars our friend they buy three shirts and three pairs of pants for a hundred dollars so that would be 3x plus 3y equals a hundred and so we have a system of equations here just like we did before and we're going to solve this probably using elimination um so let's get into it so if we in in the previous problem we multiply just one equation by something to eliminate a variable so if I wanted to eliminate let's just say I wanted to eliminate the X's well my common uh factor between or not common factor my common value between 3x and 4x is going to be 12x so if I could make that say 12x and negative 12x then I'd be in good shape so the way to make 4X say 12x is just to multiply by 3. and then to make 3x a negative 12x I'll multiply that by negative 4. so that'll work so I'm going to multiply everything in that top equation by 3 so I have 12x plus 15 y equals that'll be 450. negative 12x minus 12y equals negative 400 and at that point I can go ahead and add them together my x's will go to zero 15y minus 12y is going to be 3y and 450 minus 400 that will be just 50. and so Y is going to be 50 divided by 3 which I believe let's see 50 divided by 3 that's going to be about 16.6 repeating so we'll say 16.67 so remember what does y represent that represents cost per pair of pants so now we're going to take 16.67 and we're going to plug it back into one of the four equations we have so let's see uh let's plug it into that one there so we have 3x plus 3 times 16.67 equals 100. so 3 times 16.67 that is 50.01 it's equal to 100 we'll subtract that and we get 3x equals 49.99 and then we're going to divide that by 3. and we get x equals 16.66 so we find out that at this particular store payments and Assurance cost basically the same 1666 and 16 67. all right number 13. equation of a parabola it has a vertex of 3 4. and passes through the points 1 negative 12 and 4 0. our so let's see I know my vertex form of a parabola is a times x minus H squared plus K they want my final answer in standard form so I'll have to change that at the end but we can work for that at the end so let's see we have y equals we don't know a but we have x minus 3 squared because that's my vertex Plus 4. and then I can take either of those two points and substitute them in for X and Y so I'm going to take 4 0. and plug it in so Y is going to be 0. I have a then I have 4 minus 3 squared plus four all right so let's solve we want to find out what a is so 4 minus 3 well that's going to be one so I have 0 equals a times 1 squared plus 4 so that's zero equals one a plus four when I subtract 4 from both sides I get that a is negative 4. so my equation right now is y equals not negative 4 times x minus H squared not h x minus 3 squared plus 4. all right so that's in vertex form um I'd want to verify that when I plug in 1 I'm going to get negative 12 but we'll do that after we get standard form so how do we get standard form from here well we're going to distribute that out distribute that out so we have y equals negative four x minus 3 x minus 3 plus 4. so we have keep working here so we got negative four that's going to be x squared minus 6X plus 9 if I do the distributive property plus four so now I'll distribute negative four so I have negative 4 x squared plus 24x minus 36 plus 4 and I have y equals negative 4x squared plus 24x minus 32 and that's going to be my equation of the parabola that meets those conditions now I would want to plug in 1 here for X just to see make sure that it gives me negative 12. um so negative 4 times 1 squared plus 24 times 1 minus 32. so I have negative 4 Plus 24 so that's 20 minus 32 that's going to give me negative 12. and so I am good there all right we've got three questions left so number 14 your school wants to maximize their profit from the sales of tickets to the homecoming football game they determine the function that models The Profit they can earn in hundreds of dollars in terms of price per ticket in dollars the function is given below what price per ticket maximizes your school's profit okay so if I want to find the maximum profit remember this is a quadratic equation so the graph is going to actually open upside down something like that and where my profit is going to be my Max profit is going to be up here and that's the vertex so how do I get the vertex from standard form and I actually don't even need the whole vertex because I just need the price per ticket and since the ticket price or the yeah since the ticket price is going to be t I want to find the T coordinate of the vertex so how do I find the H or the x coordinate or the T coordinate in this case of the vertex well if you remember H has a formula it's the negative B divided by 2A and so my b in this case is 210 so I'll make it negative divided by 2 times negative 30. so I'll have negative 210 divided by negative 60. so let's do that calculator and we're going to get 3.5 so that would be three dollars and 50 cents per ticket it's going to maximize our profit all right cool number 15. right in the equation of the line that goes through the point negative 2 4 and it's perpendicular to the line Y equals two over nine X minus five so okay so we need the slope from this equation so the original slope is 2 9. the perpendicular slope is going to be remember the opposite reciprocal so opposite sign reciprocal is nine over two and now we're going to use our point-slope form which is y minus y1 equals the slope times x minus X1 and so we will have let's see this is our X1 and our y1 so we have y minus 4 equals negative nine over two times x minus negative 2. so if we simplify this a little bit we have y minus 4 equals negative nine over two times X plus two up here and I'm going to distribute that negative 9 over 2. so I have y minus 4 equals negative 9 over 2X and then minus the twos end up multiplying out so I just get minus 9. and I'm going to add 4 to both sides I'm going to have y equals negative 9 over 2x minus 5 and that's my equation that is perpendicular to that original line because it has the opposite reciprocal slope and it goes through that point negative 2 4. all right last question of this review okay so 16 we know they want us to describe the end behavior and then describe what Y is doing okay so this is going to be an exponential function and so let's just see what this would look like on a graph if we were to let's say plug in zero and see what Y is we have negative two-thirds times seven ninths to the zero power well we know anything is zero power is one so that's just going to be negative two-thirds so at zero we're at negative two thirds so just say that's in there and then let's see what happens if we put well let's just do two just kind of see where the graph's going so we have negative two-thirds 7 9 to the power of two so seven divided by nine and then we're going to raise that to the power of two I'm just going to make this a decimal so that's negative point five one eight five repeating okay so that's still going to be negative so really what this function is doing is like this so let's describe the end behavior so the end behavior as X goes to the left and as X goes to the right negative Infinity positive Infinity as X goes to the left the Y values are going down so we're going to negative Infinity as X goes to the right the Y values are going closer to zero so remember your end behavior for exponential functions um we're approaching zero all right so then let's describe what Y is doing over the whole function so the values of Y they're starting out negative and they're getting more and more negative they're getting closer to zero so if I start out at like negative two thirds and I have negative 0.5185 they're kind of increasing right because the numbers are getting bigger so Y is increasing closer to zero