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Question Video: Transformations of Functions Mathematics • 10th Grade

The function π¦ = π(π₯) is stretched in the horizontal direction by a scale factor of 1/3 and in the vertical direction by a scale factor of 1/3. 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 one-third and in the vertical direction by a scale factor of one-third. Write, in terms of π of π₯, the equation of the transformed function.

In this question, weβre given the curve of a function π of π₯ and weβre told that this is stretched in the horizontal direction by a scale factor of one-third. And then itβs also stretched in the vertical direction by a scale factor of, once again, one-third. We need to use this information to find the equation of the transformed function in terms of our original function π of π₯. We can see that weβre applying two stretches to our function π of π₯, one in the horizontal direction and one in the vertical direction. So, to answer this question, weβre first going to need to recall exactly how do we represent stretches of a function. And to do this, itβs often easiest to think of an example. Letβs start with π¦ is equal to π evaluated at π times π₯.

Remember, our π₯-values are our inputs, and weβre multiplying all of our inputs by a value of π. So weβre multiplying all of our input values of π₯ by a value of π, so this is going to be a stretch in the horizontal direction. However, we still need to work out the scale factor. And one way of finding the scale factor is to try an example. Letβs set our value of π equal to two and letβs let π evaluated at four be equal to 10. If π evaluated at four is equal to 10, then π evaluated at two multiplied by two is also π of four, so this is also going to be equal to 10. So, in this case, multiplying the π₯ inside of our function by a value of two means we needed to halve our input value of π₯ to reach the same output. So weβve halved all of our input values of π₯.

Remember, the input values of π₯ are on the π₯-axis, the horizontal axis. So this is a horizontal stretch by a factor of one-half. And, of course, if instead of using π is equal to two, weβd just use the value of π, then weβd be stretching in the horizontal direction by a scale factor of one over π. But remember, in this question, we also need to do a vertical stretch. And remember, the vertical position on a curve represents the output of the function. In other words, the π¦-coordinate represents the output of our function. So if we want to scale in the vertical direction, weβre going to want to multiply our outputs of the function. So π¦ is equal to π times π of π₯ should be a vertical stretch. We do need to work out the scale factor though.

In this case though, itβs a little easier. If weβre just multiplying our output values by a value of π, then weβre just multiplying the π¦-coordinates by π. Weβre not changing the input values at all. In other words, weβre just stretching our curve by a scale factor of π in the vertical direction. Weβre now ready to try and find an equation of our transformed function given to us in the question. First, the question wants us to stretch the curve π¦ is equal to π of π₯ in the horizontal direction by a scale factor of one-third. And we know exactly how to represent a horizontal stretch by a scale factor of one over π.

And in the question, we want a scale factor of one-third, so weβre going to need to set our value of π equal to three. In other words, all we need to do is multiply the input values of π₯ by three. Therefore, weβve shown to stretch the curve π¦ is equal to π of π₯ by a scale factor of one-third in the horizontal direction, we need π¦ is equal to π evaluated at three π₯.

But thatβs not the only thing this question wants us to do. We also need to stretch our curve in the vertical direction by a scale factor of one-third. And we know exactly how to do this by using our second rule. To vertically stretch a curve by a scale factor of one-third, weβre going to need to set our value of π equal to one-third. In other words, all weβre doing is multiplying the outputs of our function by one-third. So by applying this to our previously transformed function, we get the equation π¦ is equal to one-third times π evaluated at three π₯. And this is our final answer.

Therefore, we were able to show if the curve π¦ is equal to π of π₯ is stretched in the horizontal direction by a scale factor of one-third and in the vertical direction by a scale factor of one-third, then the equation of the transformed function, written in terms of π of π₯, is given by π¦ is equal to one-third multiplied by π evaluated at three π₯.

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