Function Transformations:  Horizontal and Vertical Stretches and Compressions

Welcome to the first of several videos on function transformations. This video focuses on horizontal and vertical stretches and compressions. So for comparing f of x to a function in this form, we'll be looking at how the value of a and b affect the graph of the original function. So if we have y equals a times f of x, where a is greater than 1, this will stretch the graph of f of x vertically by a factor of a. And if we have y equals a times f of x, where a is between 0 and 1, this will compress the graph of f of x vertically by a factor of a. So let's compare f of x to 1 half times f of x and 2 times f of x by completing t tables. So for f of x equals x squared, define y, which is square x. So we'd have 1 squared, 2 squared, 3 squared, and 4 squared. Let's first take a look at 2 times f of x, which would be 2 times x squared. Well, if this is x squared and we want to find 2x squared, we can just multiply 2 times these y values. So we'd have 2 times 1, 2 times 4, 2 times 9, and 2 times 16. And this would be a vertical stretch because notice for the same x coordinates, these y coordinates are much larger and therefore stretching it vertically. Now if we take a look at 1 half times f of x or 1 half x squared, we can just multiply 1 half times these y values to obtain 1 half x squared. So we'd have 1 half times 1, 1 half times 4, 1 half times 9, and 1 half times 16. And notice for the same x values, these y values are less and therefore would compress the graph vertically. So what you may have noticed is once you identify the value of a, to find the y coordinates of the transformed function with the same x coordinates, which would be just to multiply each y coordinate by a and leave the x coordinates the same. If we take a look at these three graphs on the same coordinate plane, we'll notice the original function is in blue. And if we stretch the blue function vertically, by a factor of 2, we would have h of x which is equal to 2 times f of x. Versus if we take the blue function and compress it vertically by a factor of 1 half, we would have g of x which is equal to 1 half times f of x. Let's take a look at an animation of this. So if a is greater than 1, we can see a vertical stretch by a factor of a. Let's take a look at what happens when a is between 0 and 1. So here we have a equals 1, a equals 0.9, 0.8 and so on. So we have a vertical compression, in this case, by a factor of 3 tenths and so on. Let's now take a look at a horizontal stretch and horizontal compression. If we have f of bx where b is greater than 1, this will compress the graph of f of x horizontally. And if we have y equals f of bx where b is between 0 and 1, this will stretch the graph of f of x horizontally. So let's go ahead and compare t tables for f of x, f of 2x and f of 1 half x. So we already know this would be 1, 4, 9 and 16. Notice now that we're multiplying x by 2 and then squaring it. So if x is 1 half, 2 times 1 half would be 1 and then square it, we'd have 1. Then 2 times 1 square would be 4, 2 times 3 half square would be 3 square root of 9 and 2 times 2 square would be 16. Notice when b is equal to 2, we leave the y coordinates the same and then either multiply the original x coordinates by 1 over b or just divide by b. Let's now take a look at f of 1 half x which would equal 1 half x squared. So b is equal to 1 half. Well, multiplying by 1 over 1 half would be the same as multiplying by 2. So in this case, if we multiply the original x coordinates by 2, we are going to get the same y coordinates. Let's take a look at the first couple. If x is equal to 2, 1 half times 2 is 1 squared, that'd be 1. 1 half times 4 would be 2 squared, that'd be 4, 9, 16. And this would be considered a horizontal stretch. If we take a look at these three graphs, again we have the original function in blue and when b is equal to 2, we have a horizontal compression so we're compressing it this way to obtain the red function f of 2x. And if we stretch the blue function horizontally when b is equal to 1 half, we would have the green function. Let's take a look at an animation of this as well. So when b is greater than 1, we have a horizontal compression as we see here. And when b is between 0 and 1, we have a horizontal stretch as we see here. Okay, let's go ahead and take a look at some of our own examples. The first thing we need to be able to do is recognize the parent function for f of x. If f of x is equal to 3 times the absolute value of x, let's let our parent function g of x equal the absolute value of x. And we should be able to make a nice graph of this without using our graphing calculator. We should all know this forms a v shape very similar to this graph. Let's go ahead and identify some key points on this graph like the origin, .22, and how about the .-22. Now looking at the original function, we should be able to make the connection that f of x is equal to 3 times g of x, which means a is equal to 3, which means that our graph is going to be a graph that's vertically stretched by a factor of 3 from this original red graph. So to find points on this transformed function, we can just multiply each of these y coordinates by 3. Since 2 times 3 would be 6, one point on the transformed graph would be negative 2 6. 0 times 3 would still be 0, so we have the origin. And here we have another y coordinate of 2, so this would become the .26. And so very quickly and easily, we can graph the transformed function based upon determining the value of a. Let's take a look at this one now. We have f of x equals the square root of 2x. Again, we should be able to recognize that the parent function, which we'll call g of x is equal to the square root of x. Let's go ahead and graph that function. If x is 0, y would be 0. If x is 1, y is 1. If x is 4, the square root of 4 would be 2. x is 9, we have the square root of 9, that'd be 3. 16. So this is the parent function, and what we're going to do is graph the transformed function. Let's go ahead and make a t-table for some of the key points on this graph, like 0, 0, 1, 1, 4, 2, 9, 3, and 16, 4. Now we need to be able to recognize that f of x is equal to g of 2x, which is equal to the square root of 2x. So that tells us that b is equal to 2. So since b is equal to 2, we have a horizontal compression, which means to find points on the transformed function, we can just multiply the x-coordinate by 1 over b, or just divide by b. So if b is equal to 2, we'll just take each of these x-coordinates, divide them by 2, and keep the y-coordinates the same. So the origin would still be on the transformed function. The next point would be 1 half 1. 4 divided by 2 would be 2, so we have the point 2, 2. 9 divided by 2 would be 9 halves of 4.5, 3, and then instead of 16, 4, we'll have 16 divided by 2, that'd be 8, 4. This should be enough points to make a nice graph of the transformed function in green. So this has been horizontally compressed, meaning pushed back in this direction. Okay, I hope you found this video helpful. Thank you for watching.