# Question Video: Transformations of Functions Mathematics

The function π¦ = π(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 π(π₯), the equation of the transformed function.

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### Video Transcript

The function π¦ is equal to π 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 π of π₯, the equation of the transformed function.

In this question, weβre given a function π of π₯. And weβre told that this is stretched in the horizontal direction by a scale factor of two. And weβre also told that itβ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 π of π₯. To answer this question, weβre first going to need to recall exactly how we represent horizontal stretches and vertical stretches in terms of functions. Letβs start by considering the curve of the function π¦ is equal to π of π₯.

If we multiply the entire π of π₯ by some constant we call π, then because π of π₯ is the outputs of our function or the π¦-coordinates of our function, weβre multiplying all of our π¦-coordinates by π. And remember, the π¦-coordinates are the vertical axis, so weβre stretching this in the vertical direction by a factor of π. And we can do something very similar for horizontal stretches.

To stretch a function in the horizontal direction, weβre going to want to multiply all of our input values of π₯. However, we do need to be very careful with this because weβre not actually stretching by a factor of π here. To see this, if we have π 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; weβve actually halved our inputs.

And another way of saying this is weβve stretched our function horizontally by a factor of one over π. And of course, we also know we can represent horizontal and vertical translations. However, we donβt need this to answer this question because weβre only stretching in the horizontal and vertical directions. Letβs now use this to answer our question.

First, we need to start with π¦ is equal to π of π₯. The first thing the question wants us to do is stretch this function horizontally by a scale factor of two. And weβve seen to stretch our function horizontally by a factor of one over π, we need to multiply all of our inputs by π. So, if we want to stretch our function horizontally by a factor of two, then we need one over π to be equal to two. And multiplying this equation through by π and dividing through by two, we see this gives us π is equal to one-half. Therefore, to stretch the function π¦ is equal to π 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 π¦ is equal to π 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 π in the vertical direction, we just multiply all of our outputs by π. So, all we need to do is multiply the function we had before by two. We get π¦ is equal to two times π of one-half π₯. And we could leave our answer like this. However, weβ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 π¦ is equal to π 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 π¦ is equal to two times π evaluated at π₯ over two.

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