Question Video: Transformations of Functions | Nagwa Question Video: Transformations of Functions | Nagwa

# Question Video: Transformations of Functions Mathematics • Second Year of Secondary School

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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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