Video: Transformations of Functions

The function 𝑦 = 𝑓(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 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 one-third and in the vertical direction by a scale factor of one-third. Write, in terms of 𝑓 of π‘₯, the equation of the transformed function.

In this question, we’re given the curve of a function 𝑓 of π‘₯ and we’re told that this is stretched in the horizontal direction by a scale factor of one-third. And then it’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 𝑓 of π‘₯. We can see that we’re applying two stretches to our function 𝑓 of π‘₯, one in the horizontal direction and one in the vertical direction. So, to answer this question, we’re first going to need to recall exactly how do we represent stretches of a function. And to do this, it’s often easiest to think of an example. Let’s start with 𝑦 is equal to 𝑔 evaluated at 𝑏 times π‘₯.

Remember, our π‘₯-values are our inputs, and we’re multiplying all of our inputs by a value of 𝑏. So we’re multiplying all of our input values of π‘₯ by a value of 𝑏, 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. Let’s set our value of 𝑏 equal to two and let’s let 𝑔 evaluated at four be equal to 10. If 𝑔 evaluated at four is equal to 10, then 𝑔 evaluated at two multiplied by two is also 𝑔 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 we’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 𝑏 is equal to two, we’d just use the value of 𝑏, then we’d be stretching in the horizontal direction by a scale factor of one over 𝑏. 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 𝑦-coordinate represents the output of our function. So if we want to scale in the vertical direction, we’re going to want to multiply our outputs of the function. So 𝑦 is equal to 𝑏 times 𝑔 of π‘₯ should be a vertical stretch. We do need to work out the scale factor though.

In this case though, it’s a little easier. If we’re just multiplying our output values by a value of 𝑏, then we’re just multiplying the 𝑦-coordinates by 𝑏. We’re not changing the input values at all. In other words, we’re just stretching our curve by a scale factor of 𝑏 in the vertical direction. We’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 𝑦 is equal to 𝑓 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 𝑏.

And in the question, we want a scale factor of one-third, so we’re going to need to set our value of 𝑏 equal to three. In other words, all we need to do is multiply the input values of π‘₯ by three. Therefore, we’ve shown to stretch the curve 𝑦 is equal to 𝑓 of π‘₯ by a scale factor of one-third in the horizontal direction, we need 𝑦 is equal to 𝑓 evaluated at three π‘₯.

But that’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, we’re going to need to set our value of 𝑏 equal to one-third. In other words, all we’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 𝑦 is equal to one-third times 𝑓 evaluated at three π‘₯. And this is our final answer.

Therefore, we were able to show if the curve 𝑦 is equal to 𝑓 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 𝑓 of π‘₯, is given by 𝑦 is equal to one-third multiplied by 𝑓 evaluated at three π‘₯.

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