Video: Transformations of Functions

The function 𝑦 = 𝑓(π‘₯) is stretched in the horizontal direction by a scale factor of 2 and in the vertical direction by a scale factor of 2. Write, in terms of 𝑓(π‘₯), the equation of the transformed function.

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Video Transcript

The function 𝑦 is equal to 𝑓 of π‘₯ is stretched in the horizontal direction by a scale factor of two and in the vertical direction by a scale factor of two. Write, in terms of 𝑓 of π‘₯, the equation of the transformed function.

In this question, we’re given a function 𝑓 of π‘₯. And we’re told that this is stretched in the horizontal direction by a scale factor of two. And we’re also told that it’s stretched in the vertical direction again by a scale factor of two. We need to find the equation of the transformed function in terms of our original function 𝑓 of π‘₯. To answer this question, we’re first going to need to recall exactly how we represent horizontal stretches and vertical stretches in terms of functions. Let’s start by considering the curve of the function 𝑦 is equal to 𝑔 of π‘₯.

If we multiply the entire 𝑔 of π‘₯ by some constant we call 𝑏, then because 𝑔 of π‘₯ is the outputs of our function or the 𝑦-coordinates of our function, we’re multiplying all of our 𝑦-coordinates by 𝑏. And remember, the 𝑦-coordinates are the vertical axis, so we’re stretching this in the vertical direction by a factor of 𝑏. And we can do something very similar for horizontal stretches.

To stretch a function in the horizontal direction, we’re going to want to multiply all of our input values of π‘₯. However, we do need to be very careful with this because we’re not actually stretching by a factor of 𝑏 here. To see this, if we have 𝑔 of one is equal to 10, then if we multiply all of our inputs of two to get the same output of 10, we now need to input a value of π‘₯ is 0.5 because, of course, two times 0.5 is equal to one. So, we have not doubled our inputs; we’ve actually halved our inputs.

And another way of saying this is we’ve stretched our function horizontally by a factor of one over 𝑏. And of course, we also know we can represent horizontal and vertical translations. However, we don’t need this to answer this question because we’re only stretching in the horizontal and vertical directions. Let’s now use this to answer our question.

First, we need to start with 𝑦 is equal to 𝑓 of π‘₯. The first thing the question wants us to do is stretch this function horizontally by a scale factor of two. And we’ve seen to stretch our function horizontally by a factor of one over 𝑏, we need to multiply all of our inputs by 𝑏. So, if we want to stretch our function horizontally by a factor of two, then we need one over 𝑏 to be equal to two. And multiplying this equation through by 𝑏 and dividing through by two, we see this gives us 𝑏 is equal to one-half. Therefore, to stretch the function 𝑦 is equal to 𝑓 of π‘₯ by a scale factor of two in the horizontal direction, we need to multiply all of our inputs by one-half. We get 𝑦 is equal to 𝑓 of one-half π‘₯.

Next, the question wants us to scale our function in the vertical direction also by a scale factor of two. And we see to stretch a function by a scale factor of 𝑏 in the vertical direction, we just multiply all of our outputs by 𝑏. So, all we need to do is multiply the function we had before by two. We get 𝑦 is equal to two times 𝑓 of one-half π‘₯. And we could leave our answer like this. However, we’re going to write one-half times π‘₯ as π‘₯ over two. And this gives us our final answer.

Therefore, we were able to show if the function 𝑦 is equal to 𝑓 of π‘₯ is stretched by a scale factor of two in the horizontal direction and by a scale factor of two in the vertical direction, then the transformed function can be represented as 𝑦 is equal to two times 𝑓 evaluated at π‘₯ over two.

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