### Video Transcript

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

So when weโre looking at transformations as the transformation is in the horizontal, so the ๐ฅ-direction, the change to the function and how we write it down is in the parentheses. For example, if we had ๐ of ๐ฅ plus two or ๐ of two ๐ฅ, these are both the transformations of ๐ of ๐ฅ. And theyโre gonna be in the horizontal ๐ฅ-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 thatโs because itโs a change to the input itself. If it was outside the parentheses, then it would be a change in the vertical or ๐ฆ-direction.

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

So therefore, if weโre looking to a stretch of a scale factor of two in the horizontal ๐ฅ-direction, then the equation of the transformed function will be ๐ฆ equals ๐ of and then either a half ๐ฅ or ๐ฅ over two cause as we said before, itโs the inverse of what you might think. So instead of being a scale factor of two, well weโd multiply ๐ฅ by two. Weโd in fact divide ๐ฅ by two. So that would give us ๐ฆ equals ๐ of ๐ฅ over two.

So weโve got the final answer. But what I thought Iโd do is draw a visual representation to help you understand whatโs actually happening. So weโve got ๐ฆ equals ๐ of ๐ฅ. And ๐ฆ equals ๐ of ๐ฅ over two. We can see thereโre two points that Iโve circled. For these two points, they have the same ๐ฆ-values or the same output.

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