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Question Video: Transformations of Functions Mathematics • 10th Grade

The function ๐‘ฆ = ๐‘“(๐‘ฅ) 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 ๐‘“(๐‘ฅ), the equation of the transformed function.

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Video Transcript

The function ๐‘ฆ is equal to ๐‘“ of ๐‘ฅ 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 ๐‘“ of ๐‘ฅ, the equation of the transformed function.

In this question, weโ€™re given the curve of a function ๐‘“ of ๐‘ฅ and weโ€™re told that this is stretched in the horizontal direction by a scale factor of one-third. And then itโ€™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 ๐‘“ of ๐‘ฅ. We can see that weโ€™re applying two stretches to our function ๐‘“ of ๐‘ฅ, one in the horizontal direction and one in the vertical direction. So, to answer this question, weโ€™re first going to need to recall exactly how do we represent stretches of a function. And to do this, itโ€™s often easiest to think of an example. Letโ€™s start with ๐‘ฆ is equal to ๐‘” evaluated at ๐‘ times ๐‘ฅ.

Remember, our ๐‘ฅ-values are our inputs, and weโ€™re multiplying all of our inputs by a value of ๐‘. So weโ€™re multiplying all of our input values of ๐‘ฅ by a value of ๐‘, 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. Letโ€™s set our value of ๐‘ equal to two and letโ€™s let ๐‘” evaluated at four be equal to 10. If ๐‘” evaluated at four is equal to 10, then ๐‘” evaluated at two multiplied by two is also ๐‘” of four, so this is also going to be equal to 10. So, in this case, multiplying the ๐‘ฅ inside of our function by a value of two means we needed to halve our input value of ๐‘ฅ to reach the same output. So weโ€™ve halved all of our input values of ๐‘ฅ.

Remember, the input values of ๐‘ฅ are on the ๐‘ฅ-axis, the horizontal axis. So this is a horizontal stretch by a factor of one-half. And, of course, if instead of using ๐‘ is equal to two, weโ€™d just use the value of ๐‘, then weโ€™d be stretching in the horizontal direction by a scale factor of one over ๐‘. 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 ๐‘ฆ-coordinate represents the output of our function. So if we want to scale in the vertical direction, weโ€™re going to want to multiply our outputs of the function. So ๐‘ฆ is equal to ๐‘ times ๐‘” of ๐‘ฅ should be a vertical stretch. We do need to work out the scale factor though.

In this case though, itโ€™s a little easier. If weโ€™re just multiplying our output values by a value of ๐‘, then weโ€™re just multiplying the ๐‘ฆ-coordinates by ๐‘. Weโ€™re not changing the input values at all. In other words, weโ€™re just stretching our curve by a scale factor of ๐‘ in the vertical direction. Weโ€™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 ๐‘ฆ is equal to ๐‘“ of ๐‘ฅ 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 ๐‘.

And in the question, we want a scale factor of one-third, so weโ€™re going to need to set our value of ๐‘ equal to three. In other words, all we need to do is multiply the input values of ๐‘ฅ by three. Therefore, weโ€™ve shown to stretch the curve ๐‘ฆ is equal to ๐‘“ of ๐‘ฅ by a scale factor of one-third in the horizontal direction, we need ๐‘ฆ is equal to ๐‘“ evaluated at three ๐‘ฅ.

But thatโ€™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, weโ€™re going to need to set our value of ๐‘ equal to one-third. In other words, all weโ€™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 ๐‘ฆ is equal to one-third times ๐‘“ evaluated at three ๐‘ฅ. And this is our final answer.

Therefore, we were able to show if the curve ๐‘ฆ is equal to ๐‘“ of ๐‘ฅ 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 ๐‘“ of ๐‘ฅ, is given by ๐‘ฆ is equal to one-third multiplied by ๐‘“ evaluated at three ๐‘ฅ.

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