The function 𝑦 equals 𝑓 of 𝑥 is stretched in the horizontal direction by a scale factor of two. Write in terms of 𝑓 of 𝑥 the equation of the transformed function.
So when we’re looking at transformations as the transformation is in the horizontal, so the 𝑥-direction, the change to the function and how we write it down is in the parentheses. For example, if we had 𝑓 of 𝑥 plus two or 𝑓 of two 𝑥, these are both the transformations of 𝑓 of 𝑥. And they’re gonna be in the horizontal 𝑥-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 that’s because it’s a change to the input itself. If it was outside the parentheses, then it would be a change in the vertical or 𝑦-direction.
The other thing to bear in mind when we’re looking at horizontal changes in transformations is it does the inverse of what you might imagine. So, for instance, if you have 𝑓 of 𝑥 plus two you might think, well, plus two we’re gonna add or shift two units to the right. Well this isn’t the case. We actually shift negative two units. So it would be to the left. Also, if we had 𝑓 of two 𝑥, you think okay so we’ve a stretch of scale factor of two. Well, in fact, no it’s the inverse. So instead of a scale factor of two, it’s dividing by two. So it’s a scale factor of a half.
So therefore, if we’re looking to a stretch of a scale factor of two in the horizontal 𝑥-direction, then the equation of the transformed function will be 𝑦 equals 𝑓 of and then either a half 𝑥 or 𝑥 over two cause as we said before, it’s the inverse of what you might think. So instead of being a scale factor of two, well we’d multiply 𝑥 by two. We’d in fact divide 𝑥 by two. So that would give us 𝑦 equals 𝑓 of 𝑥 over two.
So we’ve got the final answer. But what I thought I’d do is draw a visual representation to help you understand what’s actually happening. So we’ve got 𝑦 equals 𝑓 of 𝑥. And 𝑦 equals 𝑓 of 𝑥 over two. We can see there’re two points that I’ve circled. For these two points, they have the same 𝑦-values or the same output.
We can see, in the original function, the 𝑥-value was two at this point. But in the new function or the transformed function, the 𝑥-value is four. So as you can see, you need double the input to get the same output.