Understanding Transformations and Parent Functions

In this video, we're going to go over transformations, parent functions, their graphs, end behavior, domain and range, and also any horizontal and vertical asymptotes these functions may have. So let's go over the basics of transformations. Let's say the function that we have looks like this. So if that represents f of x, what is 2 f of x? 2 f of x is a vertical stretch, so therefore the graph will be twice as large. It's going to stretch vertically. Let's move this. over here now what about 1 half f of x what type of transformation do we have here so for the graph 1 half f of x it's going to shrink vertically so it's going to be shorter but it's still going to be equally as wide So, let's say if the amplitude is about 1, it's now going to be about a half. So, that's a vertical shrink. Now, what about f of 2x? What's going to happen here? So, f of 2x represents a horizontal shrink. Let's say if this sine wave has a period of about 2 pi, the period is now going to be pi. However, it's still going to be equally tall. The height is going to be the same, meaning the amplitude is still 1 and negative 1. It didn't change vertically. It simply shrinks horizontally. So now the period is pi instead of 2pi. If you continue to graph it, it's going to look like this. So what about f of 1 half x? So this is going to be a horizontal stretch. So the period of the sine wave will no longer be 2 pi, but it's going to stretch to 4 pi. But the height will still be the same. So it's going to be something like this. So that's a horizontal stretch. So now let's use a new function for f of x. So let's say f of x is the absolute value function of x. And it's going to look like this. So what is f of x plus 2? This function is a vertical shift up 2 units. So the graph is going to start at 2 on the y-axis, and it's going to have the same shape. So that's a vertical shift up 2. now what about this one f of x minus 2 this is a vertical shift down two units so the graph is going to start at negative 2 on the y axis and it's going to go in the upward direction What about f of x minus 2 within the inside of the parentheses? This is not going to be left 2, it's right 2. It's a horizontal shift. right two units. If you want to know why it's to the right, a quick and simple technique is to set the inside equal to zero and solve for x. If you do that, you'll see that x is positive 2 and that's where it starts on the x-axis. Now the next one that we're going to go over is f of x plus 2. So this is a horizontal shift left 2 units. So it's going to look like this. So now let's change the function. Let's use a new function. Let's use the square root of x function. Now, what's going to happen, what kind of transformation do we have if we put a negative in front of the f of x? So, the function is going to reflect across the x-axis. So, the positive y values will turn into negative y values. Now, what if the negative is on the inside of the parentheses? What kind of reflection do we have in this case? So, it's going to reflect over the y-axis. So, it's going to look like this. Now what about having a negative on the outside and on the inside? What's going to happen this time? So it's going to reflect over the x-axis and over the y-axis. So basically, it's going to reflect over the origin. And it's going to go towards quadrant 3. So it's going to look like that. Now what about this one? f-1 of x. What kind of transformation do we have? So this is the inverse function. The original function looks like this. And the line y equals x... is over here. The inverse function reflects across the line y equals x. So the inverse function is going to look like this. Notice that the red line and the green line are equally distant from the line y equals x. My graph is not perfect, but they should be equally distant from it. So it's going to reflect over the line y equals x. What about this one? f of x raised to the minus 1. This is the same as 1 over f of x. What's going to happen in this case? The function that we have, the red line, it's an increasing function. It's increasing at a decreasing rate. 1 over f of x is going to be the reciprocal. So it's going to be decreasing at a decreasing rate. It's going to look like that. On the left side we have small y values. So the small y values become large y values, and the larger y values become small y values. So it's a reciprocal function. So we had an increase in function, but now it's a decrease in function. So that covers the common transformations that you might see in algebra or a pre-calculus course. So now let's move on into the graphs of parent functions. What is the parent function of y equals x? So this graph, it's a straight line that passes through the origin, and it looks like this. So that's y is equal to x. Now, what do you think the domain for this function is? How can we figure it out? The domain represents the allowed x values. So, what you want to do is analyze the graph from left to right. The lowest x value that we can have. is all the way to the left it's negative infinity and the highest x value that we can have is positive infinity so if you don't have any fractions no radicals with an even index number no logarithms the domain is going to be all real numbers. There's no restrictions on x, so it's going to be from left to right, negative infinity to infinity. Whenever you have infinity, always use a parenthesis, never a bracket around it. Now what about the range? What are all the possible y values that we can have for this function? The lowest y value is negative infinity, and the highest y value is infinity because the arrow. As you can see, it goes this way it goes to the right and it goes up and it does that forever so the range is from negative infinity to infinity you got to view it from the bottom to the top for the domain you view it from left to right now what is the end behavior of the y equals x function so as X approaches positive infinity Y also approaches positive infinity. y is the same thing as f of x, so you can say f of x approaches positive infinity. So that's the right-end behavior. Now what about the left-end behavior? As x approaches negative infinity, What happens to y? So as we go to the left, the y value is decreasing. So y, or f of x, approaches negative infinity as well. So that's the end behavior for this function. Now, what about y equals x squared? What's the parent function for this graph? This graph looks like this. It's a parabola. It's also a quadratic function. Sometimes you might have equations like x squared plus 2x minus 1, but it's very similar to the graph that we have. By the way, this touches the origin. So what is the domain for this function? The domain for any polynomial function, be it, let's say, x cubed plus 4x minus 7, if you don't have any fractions, no radicals, no... logarithms, x could be anything. If you look at the left side, it can go all the way to the left to negative infinity, and all the way to the right to positive infinity. So therefore, the domain is negative infinity to infinity. Now what about the range? The lowest y value is 0, and the highest y value goes up to infinity. So that's the range. It starts from 0, and it includes 0, so we need a bracket, and it goes all the way to infinity. Now there's no horizontal or vertical asymptotes for this graph, so we don't have to worry about that. Now what about the n behavior? So as x approaches positive infinity, what happens? to y or f of x. Notice that as we go to the right, the graph goes up, so therefore y approaches positive infinity. Now what about the left-hand behavior? If you're taking calculus, you may have to express it using limits. So if you're in algebra, you could say as x approaches negative infinity, the y value goes up as we go to the left. So y or f of x approaches approaches positive infinity, because as we go to the left, the graph is going up. Now, if you want to use limits, you can express it this way. The limit, as x approaches negative infinity, f of x approaches positive infinity. So you can also express it that way, too. All you've got to do is add the word limit to it. So let's work on another example. So let's say if we have the function y is equal to negative x squared plus 3. How would you graph it? And what type of transformation do we have? So we just consider the parent function, which is x squared, and it looks like this. Now, negative x squared, the negative is on the outside, so therefore it's going to reflect over the x-axis. And so it's going to look like this. Now, we have negative x squared plus 3. So it's going to be this graph, but it's going to move 3 units up on the y-axis. So it's going to start at 3, but because it's negative x squared, it's going to face downward. So the domain for this graph is going to be negative infinity to infinity. That's the domain for any polynomial function. Now the range is different. The lowest y value is negative infinity because The arrows are facing in a downward direction. They can keep going down forever. Now, the highest y value is positive 3. So, therefore, the range, viewing it from bottom to top, it's negative infinity to 3. Now what about the n behavior? So as x approaches positive infinity, y approaches negative infinity. As you move to the right, the function is going down. let's use the limit for the left side the limit as X approaches negative infinity so what happens as we go to the left the function decreases f of x decreases to negative infinity as well and so that is it for this particular example Now what about this function, y is equal to x cubed? What's the parent function for this graph? So this graph, it looks like this. Actually, let me draw that better. So that's y is equal to x cubed. There's no restrictions on x, so the domain is negative infinity to infinity. The lowest y value is negative infinity, and the highest is positive infinity. Therefore... the range is also negative infinity to infinity. There's no asymptotes for this graph. And what do you think the n behavior is? So, as x approaches positive infinity, f of x approaches, let's see, as you go to the right, it goes up, so f of x approaches positive infinity. As we go to the left, it goes down, so as x approaches negative infinity, f of x, or y, also approaches negative infinity. So this is the left-end behavior, and this is the right-end behavior, which corresponds to this side. So how about this one? y is equal to the absolute value of x. What can we do for this function? So the absolute value of x function looks like this. It has a v-shape. The lowest x value is negative. and the highest is positive infinity. There's no restrictions on x, x can be anything. So therefore the domain is from negative infinity to infinity. The lowest y value is 0 and the highest is positive infinity. Therefore, the range includes zero and it goes up to infinity. There's no asymptotes for this graph and the right-hand behavior as X approaches positive infinity, f of X also approaches positive infinity. Now for the left-hand behavior, as X approaches negative infinity, Y approaches positive infinity. As you travel to the left, the function goes up. up so that's the left-hand behavior so how would you graph this function the absolute value of X minus 2 plus 3 So, this graph is going to shift 2 units to the right. Because it's x minus 2, if you set the inside equal to 0, you're going to get x equal 2. So, it's going to be a horizontal shift, right 2. And the plus 3 on the outside is a vertical shift. of 3. So it's going to start here and it's going to open forming a V shape with a slope of 1 on both sides. On the right side the slope is 1 but on the left side it's really negative 1 because you're going down. So as you move one unit to the right the point should increase by 1. Now, what is the domain for this function? Just like the other one, it's all real numbers. Now the range is going to change. The lowest y-value is 3, and the highest is infinity. So therefore, the range... is going to be from 3 to infinity, and it includes 3. The n behavior is the same. As x approaches positive infinity, y will also approach positive infinity. So that's the right n behavior. For the left n behavior, as x approaches negative infinity, it's still going to go up, so f of x approaches positive infinity. What about this one? The cube root of x. What is the parent function for it? So, this function, it's similar to x cubed, but it's a little different. For example, x cubed looks like this. The cube root of x... is that function in red. The domain is everything. Even though we have a radical, because the index number is odd, x could be anything. It could be a negative number or a positive number. Now, the lowest y value is negative infinity, and the highest y value is infinity, because as it goes to the right, it will continue to increase. So, therefore, the range also includes all y values. Now the end behavior, the right end behavior, as X approaches positive infinity, Y also approaches positive infinity. It's increasing at a decreasing rate. Now for the left side, as X approaches negative infinity, Y will approach negative infinity as well. And there are no asymptotes for this graph. Now the next function that we need to go over is the square root of x function. The parent function looks like this. It's an increase in function, but it increases at a decrease in rate. Now what is the domain for the function? Notice that the lowest x value is 0, and the highest is infinity. So therefore, the domain includes 0, is from 0 to infinity. The lowest y value is 0 and the highest is infinity. So the range is also 0 to infinity. Now, the end behavior, the right end behavior, as x approaches positive infinity, y approaches positive infinity. There's no left end behavior because x cannot approach negative infinity. It doesn't go all the way to the left. We only have an arrow that points to the right. So, there's a right end behavior. Since there's no arrow that points to the left, there's no left end behavior for the function. All you can do is say as x approaches 0 from the right side, y approaches 0. x can't be negative in this function. Now, graph all four of these functions. The square root of x, negative root x, square root negative x. and negative root negative x. So we know this function. This goes towards quadrant 1. Both x and y are positive in quadrant 1. Now, if we put the negative sign in front of the radical, it's going to reflect over the x-axis. So it's going to look like this. Notice that x is positive and y is negative. And that's true in quadrant 4. x is positive as you go to the right, y is negative as you go down. So the graph is going to go towards quadrant 4. Now in this case, we have a negative sign inside the radical. So it's going to reflect over the y-axis. Notice that x is negative, y is positive in quadrant 2. x is negative towards the left, y is positive as you go up. Now if we have a negative on the inside and on the outside, it's going to reflect across the origin. As you can see, x and y are both negative in quadrant 3, and that's where this function appears to be going. So let's say if you wish to graph 2 minus the square root of x plus 3. How would you do it? So if you set the inside equal to 0, you'll see that x is equal to negative 3, which means that it shifts 3 units to the left. And notice that there is a plus 2 on the outside, so it's going to shift 2 units up. So the graph is going to start at this point. Now what we need to know is what direction is it going to go. Is it going to go towards quadrant 1, quadrant 2, 3, or 4? Now we have a positive sign in front of x, but a negative in front of the radical. So, we can view X as being positive and Y as being negative, so it's going to go towards quadrant 4. It's going to go to the right and down. So therefore, the graph is going to look like this. look something like this. And that's simply a rough sketch. So now we can find the domain and range of this function. The lowest x value is negative 3 and the highest is positive infinity. So therefore the domain Starts from negative 3 and ends at infinity. Now the range, notice that as the function goes to the right, it also decreases. So the lowest y value is negative infinity and the highest is 2. So the range is therefore negative infinity to 2. There's no left-end behavior, but the right-end behavior, as x approaches positive infinity, y approaches negative infinity. The next function that we're going to go over is 1 over x. So this function has a horizontal asymptote at y equals 0, which is the x-axis. It also has a vertical asymptote at x equals 0, which is the y-axis. And so the graph for this function looks like this. So for the domain, x could be anything except the vertical asymptote. So it could be anything from negative infinity to infinity except 0. And the way you need to write it, it's negative infinity to 0, union 0 to infinity. Now, for the range, y can be anything from negative infinity to infinity except the horizontal asymptote. And y can't be 0. So therefore, the range is going to be negative infinity to 0, union 0 to infinity. Now what about the end behavior? As x approaches positive infinity, the function y approaches the horizontal asymptote so it's going to be 0 and for the left-hand behavior as X approaches negative infinity y approaches 0 and so that's it for this function now let's say if we have a transform rational function let's say if it's 1 over X minus 2 plus 3 To find the vertical asymptote, whenever you have a fraction, set the denominator to 0. And so for the vertical asymptote, it's x equals 2. Now for the horizontal asymptote, let's focus on 1 over x minus 2. Whenever you have a function that's bottom heavy, that is if the degree of the denominator is higher than that of the numerator, the horizontal asymptote is y is equal to 0. But notice that we have this plus 3 on the outside, so it's going to be 0 plus 3. So the horizontal asymptote is shifted 3 units up. So therefore, the equation for the horizontal asymptote is y is equal to 3. So to graph the function, let's start by plotting the vertical asymptote, which is at 2, and the horizontal asymptote, which is at 3. So, the function is going to look like this. Now, what is the domain and range for this function? For the domain, all you need to do is remove the vertical asymptote. X can be anything from negative infinity to infinity, except the vertical asymptote at 2. It doesn't... It never touches the vertical asymptote, so we can write it as negative infinity to 2 union 2 to infinity. Now, for the range, y can be anything from negative infinity to infinity, except the horizontal asymptote, which is at y, is equal to 3. So, we can write the range as negative infinity to 3 union 3 to infinity. Now what about the end behavior? What's the left end behavior and the right end behavior? The right end behavior, as x approaches positive infinity, y is going to approach the horizontal asymptote, which is 3. Now the left end behavior is the same. As x approaches negative infinity, y, or f of x, approaches the horizontal asymptote of 3. So now let's move on into another function. That's y is equal to 1 over x squared. So this function is very similar to 1 over x, but because it's squared, it's always positive, which means that it can never be below the x-axis. So the graph for 1 over x was like this, but for 1 over x squared, the left side is going to flip over the x-axis. So the function looks like this. We still have a horizontal asymptote at y equals 0 because the function is bottom heavy, and we still have a vertical asymptote at x equals 0 because if you plug in 0 for x, it will be undefined. So what is the domain and range for this function? So let's start with the domain. X could be anything from negative infinity to infinity, except the horizontal asymptote, which is at 0. So the domain is negative infinity to 0, union 0 to infinity. Now for the range, the lowest y value is 0, but it doesn't include 0 because 0 is the horizontal asymptote. But the graph goes all the way up to infinity. So therefore, the range is simply 0 to infinity. Now the end behavior, as x approaches positive infinity, y is going to approach the horizontal acetone, which is 0. And for the left end behavior, as x approaches negative infinity, y is going to approach 0. And that is it for this function. Now what about this function, y is equal to e to the x? So here we have an exponential function. How would you graph it? What's the parent function? Exponential functions have a horizontal asymptote that's y equals 0. So that's the x-axis. If there was a number like plus 1, then it would be... y is equal to 1. But since we don't have such a number, it's simply y is equal to 0. Now there's no vertical asymptote. If you make a table, if you plug in 0 for x, e to the 0 is 1. If you plug in 1, e to the first power is e, and e is about 2.7. So the first point should be somewhere around here and then the second point should be over here. So to graph it, you need to start from the horizontal asymptote and follow the two points. So that's the typical shape of an exponential function. It's an increase in function. It's a function that increases at an increase in rate. Now for the end behavior, as x approaches positive infinity, y approaches positive infinity. As x approaches negative infinity for the left-hand behavior, y approaches the horizontal asymptote, which is 0. So we have another error that goes to the left. Now what about the domain and the range? Notice that there's no restriction on x for an exponential function. So the domain is all real numbers, negative infinity to infinity. Now the range is limited. The range starts from the horizontal asymptote, and it goes to infinity. So it's 0 to infinity, but it doesn't include 0. Let's try this problem. y is equal to 2 raised to the x minus 1 plus 3. Now what I would do is make a table. I'm going to choose two points, 1 and 2. If we replace 1 into x, it's going to be 2 raised to the 1 minus 1 plus 3. 1 minus 1 is 0. Anything raised to the 0 power is 1, so 1 plus 3 is 4. Now, if we replace x with 2, what is the value of y? So, 2 raised to the 2 minus 1 plus 3. 2 to the first power is 2, so 2 plus 3 is 5. So we have those two points. Now this is an exponential function, very similar to e to the x, because x is in the exponent position. Whenever you have a variable on the exponent position, it's an exponential function. The horizontal asymptote is going to be y is equal to 3. So let's start by plotting the horizontal asymptote. And we have the point 1, 4 and 2, 5. So then the graph is going to start from the horizontal asymptote, and it's going to increase towards those two points. So, d domain for any exponential function is always R-row numbers. The range... notice that the lowest y value is the horizontal asymptote 3 and it goes all the way to infinity so the range is 3 to infinity now the right-hand behavior as x approaches positive infinity y increases towards positive infinity the left-hand behavior as x approaches negative infinity y approaches the horizontal asymptote which is 3 so that is it for exponential functions Now let's move on into logarithmic functions. Consider the function y is equal to log base 3 of x plus 1. How can we graph it? The first thing that we need to look for is the vertical asymptote. To find it, set the inside of the log function equal to 0. So therefore the vertical asymptote is x is equal to negative 1. Now, we need two points to help us graph it. To find those two points, set the inside equal to two things. One, and whatever the base is. In this case, the base is 3. And we're going to make a table. So, if x plus 1 equals 1, therefore x must be 0. 0 plus 1 is 1. If x plus 1 is 3, x must be 2 because 2 plus 1 is 3. Now what we're going to do is... We're going to take these x values and insert them into this equation. So what's log base 3 of 0 plus 1? That's basically log base 3 of 1. Log of 1 is always equal to 0. Now what's going to happen if we plug in 2? So log base 3 of 2 plus 1. 2 plus 1 is 3. Log base 3 of 3, because the numbers are the same. they're going to cancel to 1, 3 to the first power is 3. So, log base 3 of 3 is equal to 1. So, let's plot the points that we now have. So we have the first point, 0, 0, and 2, 1. So the graph is going to start from the vertical asymptote, and it's going to follow the two points. And so that's the function that we're going to get. It looks like that. So what is the domain for the log function? Notice that the lowest x value is negative 1 and the highest is infinity. Therefore, the domain is going to start from negative 1 but not including negative 1. to infinity. For the range, y could be anything for a log function. The lowest y value is negative infinity, and it keeps on going up to positive infinity. So the range is all y values. Now what about the n behavior? The right n behavior is as x approaches positive infinity, y approaches positive infinity because it's an increase in function. Now, x will never approach... negative infinity because you can never have a negative number inside a log and log you can't have a zero inside log 2 because that's the vertical asymptote so therefore X can only approach the right side of the vertical asymptote. So we can write it like this for the left-end behavior. As X approaches the vertical asymptote, which is 1, from the left side, not from the left side, from the right side. As x approaches 1 from the right side, y approaches negative infinity. So notice that y decreases as we get close to 1 from the right side. So as we travel to the x value of 1, this function decreases to negative infinity. So that's what this means. let's try this problem let's say that y is equal to 2 plus natural log x minus 3 how can we graph it so let's set the inside equal to 0 to find the vertical asymptote so therefore the vertical asymptote is x is equal to 3 Next, let's set the inside equal to two things. We're going to set it equal to 1 and whatever the base is. The base of natural log is e. So, solving for x, if we add 3 to both sides, 3 plus 1 is 4, and then the other one is simply 3. Now, what's going to happen if we plug in 4 into the equation? What is the value of y? So, 2 plus natural log of 4 minus 3. 4 minus 3. minus 3 is 1. The log or natural log of 1 is always 0. So it's 0 plus 2, which is 2. Now let's substitute X with 3 plus E. So 2 plus ln 3 plus E minus 3. The 3's will cancel, so it's simply ln E plus 2. ln E is always equal to 1, so it's 2 plus 1, which is 3. So now let's plot the points that we have. 4,2 is somewhere over here. Now, 3 plus e, where is that located? e is about 2.7, so 3 plus 2.7 is about 5.7. So it's 5.7,3, which should be somewhere over here. Now, if we connect those points with a graph, we're going to start from the vertical asymptote, and then follow the two functions, I mean, the two points. And so it should look like that. So the right-hand behavior, as x approaches positive infinity, y also approaches positive infinity. Now for the left-hand behavior, we can't go past or to the left side of the vertical asymptote. The furthest that we can go to the left side is the vertical asymptote of 3. So as x approaches 3 from the right side, y approaches negative infinity. So as we approach the vertical asymptote of 3, the function approaches all the way down to negative infinity. That's what this means. So now that we are finished with the n behavior, let's focus on the domain and a range. So what is the domain and range for this function? Notice that the lowest x value is negative 3, and the highest is infinity. Therefore, the domain is from 3 to infinity. Now for the range, the lowest y-value is negative infinity, but the highest is positive infinity. So as you can see, the range for a logarithmic function includes all y-values. The domain for an exponential function includes all x-values. The exponential function and the logarithmic function, they're like inverses of each other. So like e to the x, the inverse of that is ln x. Let's go over the trigonometric functions. Let's start with sine x. So this function is basically a sine wave, which looks like this. The general equation for a sine graph is a sine bx plus c plus d. a is the amplitude, so the number in front of sine is 1. The amplitude is the distance between the center and the top peak, which is 1. So that's the amplitude. B is the number in front of X. So therefore, for this problem, B is 1. have simply 1x. The period is equal to 2 pi divided by b. So in this case, it's 2 pi over 1, and so this is going to be 2 pi. That's one full cycle. Now there's no c value in this example. C is associated with the phase shift or the horizontal shift, which we don't have, and d is the vertical shift. So we don't have a vertical shift, but let's say if we had sine x plus 2, it would shift up 2 units. Now going back to the sine wave. What is the domain and range for this function? Keep in mind, the sine wave can continue forever. It doesn't end. So therefore, x can be anything. There's no restriction on x. So the domain for a sine wave is all row numbers. The range, however, is from negative 1 to 1. It's limited based on the amplitude. so let's say if we have the function 3 sine X minus 2 this graph is shifted two units down so this is going to be the midline of the graph Now the amplitude is 3. Negative 2 plus 3 is 1, so it's going to go all the way up to 1. Negative 2 minus 3 is negative 5. So the lowest point of the graph is negative 5. The amplitude is... number in front of Sun so this is the aptitude the distance between the center and the peak is the amplitude the period is still 2 pi so we're going to break into four intervals pi private you and three parts of positive sign starts at the center and then it goes up to one negative sign would start the center and go down to negative five positive signs going to go up to one back to the middle down to negative five and then back to the middle so it's gonna look like this that's how you can graph one period of the function but I've missed all the points so I'm gonna do that again and so that's all you need to do now keep in mind the graph will continue forever it doesn't have to stop there so the domain is all X values but the range is limited by 1 and negative 5 so the range is going to be negative 5 to positive 1 now what about the end behavior it turns out that there's no end behavior because as X approach positive infinity why does it converge to any value so there's no end behavior for sine cosine or even secant cosecant or tangent or cotangent because these functions they keep repeating for sine and cosine there are no horizontal or vertical asymptotes Now, let's graph cosine. Positive cosine starts at the top, and then you can just draw a sine wave. So one period is from here to here, so this is 2 pi. The amplitude is going to vary from 1 to negative 1, much like the sine wave. So the domain is all real numbers, but the range is still... Negative 1 to 1 there's no end behaviors, and there's no asymptotes Now negative cosine it doesn't start at the top. It's going to start at the bottom So it's going to be like this So it doesn't start at positive 1 instead of starts at negative 1, but the domain and range will still be the same Now what about this graph, tangent X? Tangent has a vertical asymptote at negative pi over 2 and pi over 2. Positive tan is an increasing function and it looks like this Negative tan looks like this. It's decreasing Now this function it repeats The period for tangent and cotangent, you can find it by using the equation pi over b instead of 2pi over b. So b is the number in front of x, so b is 1. So if you add 1pi to pi over 2, the next vertical asymptote is at 3pi over 2, and this one's at negative 3pi over 2. And we're going to have the same type of shape. Now what is the range for this function? Notice that the lowest y value is negative infinity, and the highest is infinity. So there's no restrictions on y. y can be anything. So the range is therefore... negative infinity to infinity. Now what about the domain? X can be anything except the vertical asymptotes. It might be challenging to write that in interval notation because this goes on forever. So x cannot equal plus or minus n pi over 2 where n is 1, 3, 5, dot, dot, dot, like 1, 3, 5, 7, 9, and so forth. So x could be anything else except those numbers. It might be easier to write it just like that. Now notice that there's no n behavior because this pattern keeps repeating. It doesn't converge at any point. As x approaches positive infinity, y could be positive infinity or negative infinity or something in between. So we can't really write an n behavior for this particular function. Now what about the cosecant x function? How can we graph it? Cosecant is 1 over sine. So if we can graph sine, we simply need to find the reciprocal of sine. So I'm going to graph the sine wave in blue. This is a typical sine function. I'm going to graph at least two periods. Whenever the sine wave crosses the x-axis, we're going to have a vertical asymptote. So the vertical asymptote for the cosecant function is x is equal to n pi, where n includes 0 plus or minus 1 plus or minus 2 dot So, the first one is at pi, the next one is at 2 pi, and then this is 3 pi, and 4 pi. So as you can see, the vertical asymptote is at 2 pi, and then this is 3 pi, and 4 pi. will be an n pi 0 pi 1 pi 2 pi 3 pi 4 pi and the pattern will repeat there's no horizontal asymptotes for cosecant X now to graph the function Anytime the sine wave hits 1 or negative 1, cosecant is going to touch at that point. And you just need to draw the reciprocal. So you need to reflect it over the function. So it's going to look like this. And so the graph in purple is the cosecant function. So the domain for cosecant is everything except the vertical asymptotes. So therefore x... cannot equal n pi where n is defined as 0 plus minus 1 plus minus 2 so that has to be excluded from the domain so anytime you have a vertical asymptote you need to remove those x values from the domain now what about the range for the cosecant function What is it? Notice that the lowest x value is negative infinity, since this arrow goes down to negative infinity. And then it stops at negative 1. This really... nothing in between here the sine wave shouldn't be there it just helps us to graph the cosecant function and then the graph is going to start up at 1 and go to infinity so the range is therefore negative infinity to negative 1, it includes negative 1, so this should be a bracket, and then union bracket, 1 to infinity. So that's the range of the cosecant function if the amplitude is 1. So now let's move on into the inverse functions. How would you graph the inverse sine function? So, if you graph sine, it looks like this. And whenever you want to graph the inverse function, x changes with y. You have to switch x and y. And so it would appear as if the inverse function should be like this. But notice that this is not a function because it doesn't pass the vertical line test. The inverse sine function is restricted. And so... The correct graph for the inverse sine function looks more like this. Now, for the regular sine graph, the y value would range from negative 1 to 1. So this time, the x value is going to range from negative 1 to 1. And the angles, like pi over 2... Instead of being on the x-axis, it's going to be on the y-axis. So this is going to be pi over 2. And here we're going to have negative pi over 2. Sine of 0 is 0. Sine of pi over 2 is 1. sine negative pi over 2 is negative 1. So the sine function looks like this. That's the inverse sine function. And notice that we have a closed circle. So there's really no n behavior that we can write for this function. since it doesn't goes on forever and there's no asymptotes that we have to worry about for this function but we could write the domain and the range the domain for the inverse sine function is the range for the sine function The range of sine is negative 1 to 1, so that's the domain for the inverse sine function. Now, the range of the inverse sine function is limited. It ranges from negative pi over 2 to pi over 2. It doesn't go on forever, otherwise it wouldn't be a function. So now, let's graph the inverse cosine function. So on the x-axis, it's going to be negative 1 and 1, just like before. Now, inverse sine varies between negative pi over 2 and pi over 2. It can only exist in quadrants 1 and 4. But for inverse cosine... it varies between 0 and pi. It exists in quadrants 1 and 2. Now, cosine of 0 degrees is equal to 1. have this point. Cosine pi over 2 is 0, and cosine pi is negative 1. So therefore, the graph looks something like this. So that's the inverse cosine function. The domain is negative 1 to 1. For the range, we can see that the lowest y value is 0, and the highest is pi. So the range is 0 to pi. There's no asymptotes or endpoints. behavior that we can write for this particular function. Now the last thing that we need to go over is the inverse tangent function. Now keep in mind the regular tangent function looks like this. It's an increase in function and it's about by the asymptotes pi over 2 and negative pi over 2 but it goes on from negative infinity to infinity on the y-axis so to graph inverse tangent We're going to have a horizontal asymptote instead of a vertical asymptote. So the horizontal asymptote will be at pi over 2 and negative pi over 2. want the equation of the horizontal asymptote, you can simply write y is equal to plus or minus pi over 2. Now the graph is still going to be an increase in function, and it's going to increase towards pi over 2. So what is the domain and range for this function? The lowest x value is negative infinity, and the highest is infinity. So the domain is from negative infinity to infinity. For inverse tangent, it doesn't matter what value of x you plug. in if you plug in inverse tangent of a million you're going to get about eighty nine point nine nine nine degrees it's going to be close to 90 if you plug in inverse tangent of a billion it's going to be close to 90 therefore the right end behavior as X approach positive infinity, y approaches pi over 2. Now for the left-hand behavior, if you plug in negative a million, you'll see that y approaches negative 90 or negative pi over 2. So that's the right and left-hand behavior. Now what about the range? Notice that the lowest y value is the bottom part of the horizontal asymptote, and this is the highest. So therefore, the range is from the first asymptote to the second. So it's a negative pi over 2 to pi pi over 2. But it doesn't include pi over 2 because it never actually touches the horizontal asymptote. It simply gets very close to it. And so that is it for this particular function. And that is it for this video. So thanks for watching and have a great day.

2 f of x is a vertical stretch, so therefore the graph will be twice as large. It's going to stretch vertically. Let's move this. over here now what about 1 half f of x what type of transformation do we have here so for the graph 1 half f of x it's going to shrink vertically so it's going to be shorter but it's still going to be equally as wide So, let's say if the amplitude is about 1, it's now going to be about a half. So, that's a vertical shrink.

Now, what about f of 2x? What's going to happen here? So, f of 2x represents a horizontal shrink. Let's say if this sine wave has a period of about 2 pi, the period is now going to be pi. However, it's still going to be equally tall.

The height is going to be the same, meaning the amplitude is still 1 and negative 1. It didn't change vertically. It simply shrinks horizontally. So now the period is pi instead of 2pi. If you continue to graph it, it's going to look like this.

So what about f of 1 half x? So this is going to be a horizontal stretch. So the period of the sine wave will no longer be 2 pi, but it's going to stretch to 4 pi.

But the height will still be the same. So it's going to be something like this. So that's a horizontal stretch. So now let's use a new function for f of x. So let's say f of x is the absolute value function of x.

And it's going to look like this. So what is f of x plus 2? This function is a vertical shift up 2 units. So the graph is going to start at 2 on the y-axis, and it's going to have the same shape. So that's a vertical shift up 2. now what about this one f of x minus 2 this is a vertical shift down two units so the graph is going to start at negative 2 on the y axis and it's going to go in the upward direction What about f of x minus 2 within the inside of the parentheses?

This is not going to be left 2, it's right 2. It's a horizontal shift. right two units. If you want to know why it's to the right, a quick and simple technique is to set the inside equal to zero and solve for x.

If you do that, you'll see that x is positive 2 and that's where it starts on the x-axis. Now the next one that we're going to go over is f of x plus 2. So this is a horizontal shift left 2 units. So it's going to look like this.

So now let's change the function. Let's use a new function. Let's use the square root of x function. Now, what's going to happen, what kind of transformation do we have if we put a negative in front of the f of x? So, the function is going to reflect across the x-axis.

So, the positive y values will turn into negative y values. Now, what if the negative is on the inside of the parentheses? What kind of reflection do we have in this case?

So, it's going to reflect over the y-axis. So, it's going to look like this. Now what about having a negative on the outside and on the inside? What's going to happen this time?

So it's going to reflect over the x-axis and over the y-axis. So basically, it's going to reflect over the origin. And it's going to go towards quadrant 3. So it's going to look like that.

Now what about this one? f-1 of x. What kind of transformation do we have? So this is the inverse function.

The original function looks like this. And the line y equals x... is over here. The inverse function reflects across the line y equals x. So the inverse function is going to look like this.

Notice that the red line and the green line are equally distant from the line y equals x. My graph is not perfect, but they should be equally distant from it. So it's going to reflect over the line y equals x. What about this one? f of x raised to the minus 1. This is the same as 1 over f of x.

What's going to happen in this case? The function that we have, the red line, it's an increasing function. It's increasing at a decreasing rate. 1 over f of x is going to be the reciprocal. So it's going to be decreasing at a decreasing rate.

It's going to look like that. On the left side we have small y values. So the small y values become large y values, and the larger y values become small y values. So it's a reciprocal function.

So we had an increase in function, but now it's a decrease in function. So that covers the common transformations that you might see in algebra or a pre-calculus course. So now let's move on into the graphs of parent functions. What is the parent function of y equals x?

So this graph, it's a straight line that passes through the origin, and it looks like this. So that's y is equal to x. Now, what do you think the domain for this function is? How can we figure it out?

The domain represents the allowed x values. So, what you want to do is analyze the graph from left to right. The lowest x value that we can have.

is all the way to the left it's negative infinity and the highest x value that we can have is positive infinity so if you don't have any fractions no radicals with an even index number no logarithms the domain is going to be all real numbers. There's no restrictions on x, so it's going to be from left to right, negative infinity to infinity. Whenever you have infinity, always use a parenthesis, never a bracket around it. Now what about the range?

What are all the possible y values that we can have for this function? The lowest y value is negative infinity, and the highest y value is infinity because the arrow. As you can see, it goes this way it goes to the right and it goes up and it does that forever so the range is from negative infinity to infinity you got to view it from the bottom to the top for the domain you view it from left to right now what is the end behavior of the y equals x function so as X approaches positive infinity Y also approaches positive infinity.

y is the same thing as f of x, so you can say f of x approaches positive infinity. So that's the right-end behavior. Now what about the left-end behavior?

As x approaches negative infinity, What happens to y? So as we go to the left, the y value is decreasing. So y, or f of x, approaches negative infinity as well. So that's the end behavior for this function. Now, what about y equals x squared?

What's the parent function for this graph? This graph looks like this. It's a parabola. It's also a quadratic function.

Sometimes you might have equations like x squared plus 2x minus 1, but it's very similar to the graph that we have. By the way, this touches the origin. So what is the domain for this function? The domain for any polynomial function, be it, let's say, x cubed plus 4x minus 7, if you don't have any fractions, no radicals, no...

logarithms, x could be anything. If you look at the left side, it can go all the way to the left to negative infinity, and all the way to the right to positive infinity. So therefore, the domain is negative infinity to infinity. Now what about the range? The lowest y value is 0, and the highest y value goes up to infinity.

So that's the range. It starts from 0, and it includes 0, so we need a bracket, and it goes all the way to infinity. Now there's no horizontal or vertical asymptotes for this graph, so we don't have to worry about that.

Now what about the n behavior? So as x approaches positive infinity, what happens? to y or f of x.

Notice that as we go to the right, the graph goes up, so therefore y approaches positive infinity. Now what about the left-hand behavior? If you're taking calculus, you may have to express it using limits.

So if you're in algebra, you could say as x approaches negative infinity, the y value goes up as we go to the left. So y or f of x approaches approaches positive infinity, because as we go to the left, the graph is going up. Now, if you want to use limits, you can express it this way. The limit, as x approaches negative infinity, f of x approaches positive infinity. So you can also express it that way, too.

All you've got to do is add the word limit to it. So let's work on another example. So let's say if we have the function y is equal to negative x squared plus 3. How would you graph it?

And what type of transformation do we have? So we just consider the parent function, which is x squared, and it looks like this. Now, negative x squared, the negative is on the outside, so therefore it's going to reflect over the x-axis. And so it's going to look like this. Now, we have negative x squared plus 3. So it's going to be this graph, but it's going to move 3 units up on the y-axis.

So it's going to start at 3, but because it's negative x squared, it's going to face downward. So the domain for this graph is going to be negative infinity to infinity. That's the domain for any polynomial function.

Now the range is different. The lowest y value is negative infinity because The arrows are facing in a downward direction. They can keep going down forever.

Now, the highest y value is positive 3. So, therefore, the range, viewing it from bottom to top, it's negative infinity to 3. Now what about the n behavior? So as x approaches positive infinity, y approaches negative infinity. As you move to the right, the function is going down.

let's use the limit for the left side the limit as X approaches negative infinity so what happens as we go to the left the function decreases f of x decreases to negative infinity as well and so that is it for this particular example Now what about this function, y is equal to x cubed? What's the parent function for this graph? So this graph, it looks like this. Actually, let me draw that better. So that's y is equal to x cubed.

There's no restrictions on x, so the domain is negative infinity to infinity. The lowest y value is negative infinity, and the highest is positive infinity. Therefore... the range is also negative infinity to infinity. There's no asymptotes for this graph.

And what do you think the n behavior is? So, as x approaches positive infinity, f of x approaches, let's see, as you go to the right, it goes up, so f of x approaches positive infinity. As we go to the left, it goes down, so as x approaches negative infinity, f of x, or y, also approaches negative infinity. So this is the left-end behavior, and this is the right-end behavior, which corresponds to this side.

So how about this one? y is equal to the absolute value of x. What can we do for this function?

So the absolute value of x function looks like this. It has a v-shape. The lowest x value is negative.

and the highest is positive infinity. There's no restrictions on x, x can be anything. So therefore the domain is from negative infinity to infinity.

The lowest y value is 0 and the highest is positive infinity. Therefore, the range includes zero and it goes up to infinity. There's no asymptotes for this graph and the right-hand behavior as X approaches positive infinity, f of X also approaches positive infinity.

Now for the left-hand behavior, as X approaches negative infinity, Y approaches positive infinity. As you travel to the left, the function goes up. up so that's the left-hand behavior so how would you graph this function the absolute value of X minus 2 plus 3 So, this graph is going to shift 2 units to the right.

Because it's x minus 2, if you set the inside equal to 0, you're going to get x equal 2. So, it's going to be a horizontal shift, right 2. And the plus 3 on the outside is a vertical shift. of 3. So it's going to start here and it's going to open forming a V shape with a slope of 1 on both sides. On the right side the slope is 1 but on the left side it's really negative 1 because you're going down. So as you move one unit to the right the point should increase by 1. Now, what is the domain for this function?

Just like the other one, it's all real numbers. Now the range is going to change. The lowest y-value is 3, and the highest is infinity.

So therefore, the range... is going to be from 3 to infinity, and it includes 3. The n behavior is the same. As x approaches positive infinity, y will also approach positive infinity.

So that's the right n behavior. For the left n behavior, as x approaches negative infinity, it's still going to go up, so f of x approaches positive infinity. What about this one?

The cube root of x. What is the parent function for it? So, this function, it's similar to x cubed, but it's a little different.

For example, x cubed looks like this. The cube root of x... is that function in red. The domain is everything. Even though we have a radical, because the index number is odd, x could be anything.

It could be a negative number or a positive number. Now, the lowest y value is negative infinity, and the highest y value is infinity, because as it goes to the right, it will continue to increase. So, therefore, the range also includes all y values.

Now the end behavior, the right end behavior, as X approaches positive infinity, Y also approaches positive infinity. It's increasing at a decreasing rate. Now for the left side, as X approaches negative infinity, Y will approach negative infinity as well.

And there are no asymptotes for this graph. Now the next function that we need to go over is the square root of x function. The parent function looks like this.

It's an increase in function, but it increases at a decrease in rate. Now what is the domain for the function? Notice that the lowest x value is 0, and the highest is infinity. So therefore, the domain includes 0, is from 0 to infinity.

The lowest y value is 0 and the highest is infinity. So the range is also 0 to infinity. Now, the end behavior, the right end behavior, as x approaches positive infinity, y approaches positive infinity.

There's no left end behavior because x cannot approach negative infinity. It doesn't go all the way to the left. We only have an arrow that points to the right. So, there's a right end behavior.

Since there's no arrow that points to the left, there's no left end behavior for the function. All you can do is say as x approaches 0 from the right side, y approaches 0. x can't be negative in this function. Now, graph all four of these functions. The square root of x, negative root x, square root negative x.

and negative root negative x. So we know this function. This goes towards quadrant 1. Both x and y are positive in quadrant 1. Now, if we put the negative sign in front of the radical, it's going to reflect over the x-axis. So it's going to look like this. Notice that x is positive and y is negative.

And that's true in quadrant 4. x is positive as you go to the right, y is negative as you go down. So the graph is going to go towards quadrant 4. Now in this case, we have a negative sign inside the radical. So it's going to reflect over the y-axis. Notice that x is negative, y is positive in quadrant 2. x is negative towards the left, y is positive as you go up.

Now if we have a negative on the inside and on the outside, it's going to reflect across the origin. As you can see, x and y are both negative in quadrant 3, and that's where this function appears to be going. So let's say if you wish to graph 2 minus the square root of x plus 3. How would you do it? So if you set the inside equal to 0, you'll see that x is equal to negative 3, which means that it shifts 3 units to the left. And notice that there is a plus 2 on the outside, so it's going to shift 2 units up.

So the graph is going to start at this point. Now what we need to know is what direction is it going to go. Is it going to go towards quadrant 1, quadrant 2, 3, or 4? Now we have a positive sign in front of x, but a negative in front of the radical.

So, we can view X as being positive and Y as being negative, so it's going to go towards quadrant 4. It's going to go to the right and down. So therefore, the graph is going to look like this. look something like this.

And that's simply a rough sketch. So now we can find the domain and range of this function. The lowest x value is negative 3 and the highest is positive infinity. So therefore the domain Starts from negative 3 and ends at infinity.

Now the range, notice that as the function goes to the right, it also decreases. So the lowest y value is negative infinity and the highest is 2. So the range is therefore negative infinity to 2. There's no left-end behavior, but the right-end behavior, as x approaches positive infinity, y approaches negative infinity. The next function that we're going to go over is 1 over x.

So this function has a horizontal asymptote at y equals 0, which is the x-axis. It also has a vertical asymptote at x equals 0, which is the y-axis. And so the graph for this function looks like this.

So for the domain, x could be anything except the vertical asymptote. So it could be anything from negative infinity to infinity except 0. And the way you need to write it, it's negative infinity to 0, union 0 to infinity. Now, for the range, y can be anything from negative infinity to infinity except the horizontal asymptote. And y can't be 0. So therefore, the range is going to be negative infinity to 0, union 0 to infinity. Now what about the end behavior?

As x approaches positive infinity, the function y approaches the horizontal asymptote so it's going to be 0 and for the left-hand behavior as X approaches negative infinity y approaches 0 and so that's it for this function now let's say if we have a transform rational function let's say if it's 1 over X minus 2 plus 3 To find the vertical asymptote, whenever you have a fraction, set the denominator to 0. And so for the vertical asymptote, it's x equals 2. Now for the horizontal asymptote, let's focus on 1 over x minus 2. Whenever you have a function that's bottom heavy, that is if the degree of the denominator is higher than that of the numerator, the horizontal asymptote is y is equal to 0. But notice that we have this plus 3 on the outside, so it's going to be 0 plus 3. So the horizontal asymptote is shifted 3 units up. So therefore, the equation for the horizontal asymptote is y is equal to 3. So to graph the function, let's start by plotting the vertical asymptote, which is at 2, and the horizontal asymptote, which is at 3. So, the function is going to look like this. Now, what is the domain and range for this function?

For the domain, all you need to do is remove the vertical asymptote. X can be anything from negative infinity to infinity, except the vertical asymptote at 2. It doesn't... It never touches the vertical asymptote, so we can write it as negative infinity to 2 union 2 to infinity.

Now, for the range, y can be anything from negative infinity to infinity, except the horizontal asymptote, which is at y, is equal to 3. So, we can write the range as negative infinity to 3 union 3 to infinity. Now what about the end behavior? What's the left end behavior and the right end behavior? The right end behavior, as x approaches positive infinity, y is going to approach the horizontal asymptote, which is 3. Now the left end behavior is the same. As x approaches negative infinity, y, or f of x, approaches the horizontal asymptote of 3. So now let's move on into another function.

That's y is equal to 1 over x squared. So this function is very similar to 1 over x, but because it's squared, it's always positive, which means that it can never be below the x-axis. So the graph for 1 over x was like this, but for 1 over x squared, the left side is going to flip over the x-axis.

So the function looks like this. We still have a horizontal asymptote at y equals 0 because the function is bottom heavy, and we still have a vertical asymptote at x equals 0 because if you plug in 0 for x, it will be undefined. So what is the domain and range for this function? So let's start with the domain. X could be anything from negative infinity to infinity, except the horizontal asymptote, which is at 0. So the domain is negative infinity to 0, union 0 to infinity.

Now for the range, the lowest y value is 0, but it doesn't include 0 because 0 is the horizontal asymptote. But the graph goes all the way up to infinity. So therefore, the range is simply 0 to infinity.

Now the end behavior, as x approaches positive infinity, y is going to approach the horizontal acetone, which is 0. And for the left end behavior, as x approaches negative infinity, y is going to approach 0. And that is it for this function. Now what about this function, y is equal to e to the x? So here we have an exponential function. How would you graph it?

What's the parent function? Exponential functions have a horizontal asymptote that's y equals 0. So that's the x-axis. If there was a number like plus 1, then it would be... y is equal to 1. But since we don't have such a number, it's simply y is equal to 0. Now there's no vertical asymptote.

If you make a table, if you plug in 0 for x, e to the 0 is 1. If you plug in 1, e to the first power is e, and e is about 2.7. So the first point should be somewhere around here and then the second point should be over here. So to graph it, you need to start from the horizontal asymptote and follow the two points. So that's the typical shape of an exponential function.

It's an increase in function. It's a function that increases at an increase in rate. Now for the end behavior, as x approaches positive infinity, y approaches positive infinity.

As x approaches negative infinity for the left-hand behavior, y approaches the horizontal asymptote, which is 0. So we have another error that goes to the left. Now what about the domain and the range? Notice that there's no restriction on x for an exponential function.

So the domain is all real numbers, negative infinity to infinity. Now the range is limited. The range starts from the horizontal asymptote, and it goes to infinity. So it's 0 to infinity, but it doesn't include 0. Let's try this problem.

y is equal to 2 raised to the x minus 1 plus 3. Now what I would do is make a table. I'm going to choose two points, 1 and 2. If we replace 1 into x, it's going to be 2 raised to the 1 minus 1 plus 3. 1 minus 1 is 0. Anything raised to the 0 power is 1, so 1 plus 3 is 4. Now, if we replace x with 2, what is the value of y? So, 2 raised to the 2 minus 1 plus 3. 2 to the first power is 2, so 2 plus 3 is 5. So we have those two points.

Now this is an exponential function, very similar to e to the x, because x is in the exponent position. Whenever you have a variable on the exponent position, it's an exponential function. The horizontal asymptote is going to be y is equal to 3. So let's start by plotting the horizontal asymptote. And we have the point 1, 4 and 2, 5. So then the graph is going to start from the horizontal asymptote, and it's going to increase towards those two points. So, d domain for any exponential function is always R-row numbers.

The range... notice that the lowest y value is the horizontal asymptote 3 and it goes all the way to infinity so the range is 3 to infinity now the right-hand behavior as x approaches positive infinity y increases towards positive infinity the left-hand behavior as x approaches negative infinity y approaches the horizontal asymptote which is 3 so that is it for exponential functions Now let's move on into logarithmic functions. Consider the function y is equal to log base 3 of x plus 1. How can we graph it?

The first thing that we need to look for is the vertical asymptote. To find it, set the inside of the log function equal to 0. So therefore the vertical asymptote is x is equal to negative 1. Now, we need two points to help us graph it. To find those two points, set the inside equal to two things.

One, and whatever the base is. In this case, the base is 3. And we're going to make a table. So, if x plus 1 equals 1, therefore x must be 0. 0 plus 1 is 1. If x plus 1 is 3, x must be 2 because 2 plus 1 is 3. Now what we're going to do is... We're going to take these x values and insert them into this equation.

So what's log base 3 of 0 plus 1? That's basically log base 3 of 1. Log of 1 is always equal to 0. Now what's going to happen if we plug in 2? So log base 3 of 2 plus 1. 2 plus 1 is 3. Log base 3 of 3, because the numbers are the same. they're going to cancel to 1, 3 to the first power is 3. So, log base 3 of 3 is equal to 1. So, let's plot the points that we now have. So we have the first point, 0, 0, and 2, 1. So the graph is going to start from the vertical asymptote, and it's going to follow the two points.

And so that's the function that we're going to get. It looks like that. So what is the domain for the log function? Notice that the lowest x value is negative 1 and the highest is infinity.

Therefore, the domain is going to start from negative 1 but not including negative 1. to infinity. For the range, y could be anything for a log function. The lowest y value is negative infinity, and it keeps on going up to positive infinity.

So the range is all y values. Now what about the n behavior? The right n behavior is as x approaches positive infinity, y approaches positive infinity because it's an increase in function. Now, x will never approach...

negative infinity because you can never have a negative number inside a log and log you can't have a zero inside log 2 because that's the vertical asymptote so therefore X can only approach the right side of the vertical asymptote. So we can write it like this for the left-end behavior. As X approaches the vertical asymptote, which is 1, from the left side, not from the left side, from the right side.

As x approaches 1 from the right side, y approaches negative infinity. So notice that y decreases as we get close to 1 from the right side. So as we travel to the x value of 1, this function decreases to negative infinity. So that's what this means.

let's try this problem let's say that y is equal to 2 plus natural log x minus 3 how can we graph it so let's set the inside equal to 0 to find the vertical asymptote so therefore the vertical asymptote is x is equal to 3 Next, let's set the inside equal to two things. We're going to set it equal to 1 and whatever the base is. The base of natural log is e.

So, solving for x, if we add 3 to both sides, 3 plus 1 is 4, and then the other one is simply 3. Now, what's going to happen if we plug in 4 into the equation? What is the value of y? So, 2 plus natural log of 4 minus 3. 4 minus 3. minus 3 is 1. The log or natural log of 1 is always 0. So it's 0 plus 2, which is 2. Now let's substitute X with 3 plus E. So 2 plus ln 3 plus E minus 3. The 3's will cancel, so it's simply ln E plus 2. ln E is always equal to 1, so it's 2 plus 1, which is 3. So now let's plot the points that we have.

4,2 is somewhere over here. Now, 3 plus e, where is that located? e is about 2.7, so 3 plus 2.7 is about 5.7.

So it's 5.7,3, which should be somewhere over here. Now, if we connect those points with a graph, we're going to start from the vertical asymptote, and then follow the two functions, I mean, the two points. And so it should look like that.

So the right-hand behavior, as x approaches positive infinity, y also approaches positive infinity. Now for the left-hand behavior, we can't go past or to the left side of the vertical asymptote. The furthest that we can go to the left side is the vertical asymptote of 3. So as x approaches 3 from the right side, y approaches negative infinity. So as we approach the vertical asymptote of 3, the function approaches all the way down to negative infinity. That's what this means.

So now that we are finished with the n behavior, let's focus on the domain and a range. So what is the domain and range for this function? Notice that the lowest x value is negative 3, and the highest is infinity.

Therefore, the domain is from 3 to infinity. Now for the range, the lowest y-value is negative infinity, but the highest is positive infinity. So as you can see, the range for a logarithmic function includes all y-values. The domain for an exponential function includes all x-values. The exponential function and the logarithmic function, they're like inverses of each other.

So like e to the x, the inverse of that is ln x. Let's go over the trigonometric functions. Let's start with sine x.

So this function is basically a sine wave, which looks like this. The general equation for a sine graph is a sine bx plus c plus d. a is the amplitude, so the number in front of sine is 1. The amplitude is the distance between the center and the top peak, which is 1. So that's the amplitude. B is the number in front of X. So therefore, for this problem, B is 1. have simply 1x.

The period is equal to 2 pi divided by b. So in this case, it's 2 pi over 1, and so this is going to be 2 pi. That's one full cycle. Now there's no c value in this example. C is associated with the phase shift or the horizontal shift, which we don't have, and d is the vertical shift.

So we don't have a vertical shift, but let's say if we had sine x plus 2, it would shift up 2 units. Now going back to the sine wave. What is the domain and range for this function? Keep in mind, the sine wave can continue forever. It doesn't end.

So therefore, x can be anything. There's no restriction on x. So the domain for a sine wave is all row numbers.

The range, however, is from negative 1 to 1. It's limited based on the amplitude. so let's say if we have the function 3 sine X minus 2 this graph is shifted two units down so this is going to be the midline of the graph Now the amplitude is 3. Negative 2 plus 3 is 1, so it's going to go all the way up to 1. Negative 2 minus 3 is negative 5. So the lowest point of the graph is negative 5. The amplitude is... number in front of Sun so this is the aptitude the distance between the center and the peak is the amplitude the period is still 2 pi so we're going to break into four intervals pi private you and three parts of positive sign starts at the center and then it goes up to one negative sign would start the center and go down to negative five positive signs going to go up to one back to the middle down to negative five and then back to the middle so it's gonna look like this that's how you can graph one period of the function but I've missed all the points so I'm gonna do that again and so that's all you need to do now keep in mind the graph will continue forever it doesn't have to stop there so the domain is all X values but the range is limited by 1 and negative 5 so the range is going to be negative 5 to positive 1 now what about the end behavior it turns out that there's no end behavior because as X approach positive infinity why does it converge to any value so there's no end behavior for sine cosine or even secant cosecant or tangent or cotangent because these functions they keep repeating for sine and cosine there are no horizontal or vertical asymptotes Now, let's graph cosine.

Positive cosine starts at the top, and then you can just draw a sine wave. So one period is from here to here, so this is 2 pi. The amplitude is going to vary from 1 to negative 1, much like the sine wave.

So the domain is all real numbers, but the range is still... Negative 1 to 1 there's no end behaviors, and there's no asymptotes Now negative cosine it doesn't start at the top. It's going to start at the bottom So it's going to be like this So it doesn't start at positive 1 instead of starts at negative 1, but the domain and range will still be the same Now what about this graph, tangent X? Tangent has a vertical asymptote at negative pi over 2 and pi over 2. Positive tan is an increasing function and it looks like this Negative tan looks like this. It's decreasing Now this function it repeats The period for tangent and cotangent, you can find it by using the equation pi over b instead of 2pi over b.

So b is the number in front of x, so b is 1. So if you add 1pi to pi over 2, the next vertical asymptote is at 3pi over 2, and this one's at negative 3pi over 2. And we're going to have the same type of shape. Now what is the range for this function? Notice that the lowest y value is negative infinity, and the highest is infinity. So there's no restrictions on y. y can be anything.

So the range is therefore... negative infinity to infinity. Now what about the domain? X can be anything except the vertical asymptotes. It might be challenging to write that in interval notation because this goes on forever.

So x cannot equal plus or minus n pi over 2 where n is 1, 3, 5, dot, dot, dot, like 1, 3, 5, 7, 9, and so forth. So x could be anything else except those numbers. It might be easier to write it just like that.

Now notice that there's no n behavior because this pattern keeps repeating. It doesn't converge at any point. As x approaches positive infinity, y could be positive infinity or negative infinity or something in between. So we can't really write an n behavior for this particular function. Now what about the cosecant x function?

How can we graph it? Cosecant is 1 over sine. So if we can graph sine, we simply need to find the reciprocal of sine. So I'm going to graph the sine wave in blue. This is a typical sine function.

I'm going to graph at least two periods. Whenever the sine wave crosses the x-axis, we're going to have a vertical asymptote. So the vertical asymptote for the cosecant function is x is equal to n pi, where n includes 0 plus or minus 1 plus or minus 2 dot So, the first one is at pi, the next one is at 2 pi, and then this is 3 pi, and 4 pi.

So as you can see, the vertical asymptote is at 2 pi, and then this is 3 pi, and 4 pi. will be an n pi 0 pi 1 pi 2 pi 3 pi 4 pi and the pattern will repeat there's no horizontal asymptotes for cosecant X now to graph the function Anytime the sine wave hits 1 or negative 1, cosecant is going to touch at that point. And you just need to draw the reciprocal. So you need to reflect it over the function. So it's going to look like this.

And so the graph in purple is the cosecant function. So the domain for cosecant is everything except the vertical asymptotes. So therefore x...

cannot equal n pi where n is defined as 0 plus minus 1 plus minus 2 so that has to be excluded from the domain so anytime you have a vertical asymptote you need to remove those x values from the domain now what about the range for the cosecant function What is it? Notice that the lowest x value is negative infinity, since this arrow goes down to negative infinity. And then it stops at negative 1. This really... nothing in between here the sine wave shouldn't be there it just helps us to graph the cosecant function and then the graph is going to start up at 1 and go to infinity so the range is therefore negative infinity to negative 1, it includes negative 1, so this should be a bracket, and then union bracket, 1 to infinity.

So that's the range of the cosecant function if the amplitude is 1. So now let's move on into the inverse functions. How would you graph the inverse sine function? So, if you graph sine, it looks like this.

And whenever you want to graph the inverse function, x changes with y. You have to switch x and y. And so it would appear as if the inverse function should be like this.

But notice that this is not a function because it doesn't pass the vertical line test. The inverse sine function is restricted. And so...

The correct graph for the inverse sine function looks more like this. Now, for the regular sine graph, the y value would range from negative 1 to 1. So this time, the x value is going to range from negative 1 to 1. And the angles, like pi over 2... Instead of being on the x-axis, it's going to be on the y-axis.

So this is going to be pi over 2. And here we're going to have negative pi over 2. Sine of 0 is 0. Sine of pi over 2 is 1. sine negative pi over 2 is negative 1. So the sine function looks like this. That's the inverse sine function. And notice that we have a closed circle.

So there's really no n behavior that we can write for this function. since it doesn't goes on forever and there's no asymptotes that we have to worry about for this function but we could write the domain and the range the domain for the inverse sine function is the range for the sine function The range of sine is negative 1 to 1, so that's the domain for the inverse sine function. Now, the range of the inverse sine function is limited. It ranges from negative pi over 2 to pi over 2. It doesn't go on forever, otherwise it wouldn't be a function. So now, let's graph the inverse cosine function.

So on the x-axis, it's going to be negative 1 and 1, just like before. Now, inverse sine varies between negative pi over 2 and pi over 2. It can only exist in quadrants 1 and 4. But for inverse cosine... it varies between 0 and pi. It exists in quadrants 1 and 2. Now, cosine of 0 degrees is equal to 1. have this point. Cosine pi over 2 is 0, and cosine pi is negative 1. So therefore, the graph looks something like this.

So that's the inverse cosine function. The domain is negative 1 to 1. For the range, we can see that the lowest y value is 0, and the highest is pi. So the range is 0 to pi.

There's no asymptotes or endpoints. behavior that we can write for this particular function. Now the last thing that we need to go over is the inverse tangent function.

Now keep in mind the regular tangent function looks like this. It's an increase in function and it's about by the asymptotes pi over 2 and negative pi over 2 but it goes on from negative infinity to infinity on the y-axis so to graph inverse tangent We're going to have a horizontal asymptote instead of a vertical asymptote. So the horizontal asymptote will be at pi over 2 and negative pi over 2. want the equation of the horizontal asymptote, you can simply write y is equal to plus or minus pi over 2. Now the graph is still going to be an increase in function, and it's going to increase towards pi over 2. So what is the domain and range for this function? The lowest x value is negative infinity, and the highest is infinity. So the domain is from negative infinity to infinity.

For inverse tangent, it doesn't matter what value of x you plug. in if you plug in inverse tangent of a million you're going to get about eighty nine point nine nine nine degrees it's going to be close to 90 if you plug in inverse tangent of a billion it's going to be close to 90 therefore the right end behavior as X approach positive infinity, y approaches pi over 2. Now for the left-hand behavior, if you plug in negative a million, you'll see that y approaches negative 90 or negative pi over 2. So that's the right and left-hand behavior. Now what about the range?

Notice that the lowest y value is the bottom part of the horizontal asymptote, and this is the highest. So therefore, the range is from the first asymptote to the second. So it's a negative pi over 2 to pi pi over 2. But it doesn't include pi over 2 because it never actually touches the horizontal asymptote. It simply gets very close to it.

And so that is it for this particular function. And that is it for this video. So thanks for watching and have a great day.

Transcript for:Understanding Transformations and Parent Functions

Transcript for:
Understanding Transformations and Parent Functions