Video: Transformations of Functions

The function ๐‘ฆ = ๐‘“(๐‘ฅ) is stretched in the horizontal direction by a scale factor of 2. Write, in terms of ๐‘“(๐‘ฅ), the equation of the transformed function.

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Video Transcript

The function ๐‘ฆ equals ๐‘“ of ๐‘ฅ is stretched in the horizontal direction by a scale factor of two. Write in terms of ๐‘“ of ๐‘ฅ the equation of the transformed function.

So when weโ€™re looking at transformations as the transformation is in the horizontal, so the ๐‘ฅ-direction, the change to the function and how we write it down is in the parentheses. For example, if we had ๐‘“ of ๐‘ฅ plus two or ๐‘“ of two ๐‘ฅ, these are both the transformations of ๐‘“ of ๐‘ฅ. And theyโ€™re gonna be in the horizontal ๐‘ฅ-direction. One of them is a shift. And one of them is a stretch because as you see the number two is within the parentheses. And thatโ€™s because itโ€™s a change to the input itself. If it was outside the parentheses, then it would be a change in the vertical or ๐‘ฆ-direction.

The other thing to bear in mind when weโ€™re looking at horizontal changes in transformations is it does the inverse of what you might imagine. So, for instance, if you have ๐‘“ of ๐‘ฅ plus two you might think, well, plus two weโ€™re gonna add or shift two units to the right. Well this isnโ€™t the case. We actually shift negative two units. So it would be to the left. Also, if we had ๐‘“ of two ๐‘ฅ, you think okay so weโ€™ve a stretch of scale factor of two. Well, in fact, no itโ€™s the inverse. So instead of a scale factor of two, itโ€™s dividing by two. So itโ€™s a scale factor of a half.

So therefore, if weโ€™re looking to a stretch of a scale factor of two in the horizontal ๐‘ฅ-direction, then the equation of the transformed function will be ๐‘ฆ equals ๐‘“ of and then either a half ๐‘ฅ or ๐‘ฅ over two cause as we said before, itโ€™s the inverse of what you might think. So instead of being a scale factor of two, well weโ€™d multiply ๐‘ฅ by two. Weโ€™d in fact divide ๐‘ฅ by two. So that would give us ๐‘ฆ equals ๐‘“ of ๐‘ฅ over two.

So weโ€™ve got the final answer. But what I thought Iโ€™d do is draw a visual representation to help you understand whatโ€™s actually happening. So weโ€™ve got ๐‘ฆ equals ๐‘“ of ๐‘ฅ. And ๐‘ฆ equals ๐‘“ of ๐‘ฅ over two. We can see thereโ€™re two points that Iโ€™ve circled. For these two points, they have the same ๐‘ฆ-values or the same output.

We can see, in the original function, the ๐‘ฅ-value was two at this point. But in the new function or the transformed function, the ๐‘ฅ-value is four. So as you can see, you need double the input to get the same output.

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