Question Video: Combined Transformations of Graphs Mathematics

The graph of the function 𝑓 is produced from that of 𝑔(π‘₯) = π‘₯Β² by the following steps: a shift of 4 to the right, a dilation by a scale factor of 1/4, and a shift of 4 up. What is the function 𝑓?


Video Transcript

The graph of the function 𝑓 is produced from that of 𝑔 of π‘₯ equals π‘₯ squared by the following steps: a shift of four to the right, a dilation by a scale factor of one-fourth, and a shift of four up. What is the function 𝑓?

When a quadratic is in vertex form, it’s easy to see how it can be transformed or shifted. π‘Ž, β„Ž, and π‘˜ represent different things that we could do to our graph to change it or move it.

π‘Ž represents the scale factor. It decides how thin or wide the graph will be. β„Ž will shift our graph left and right. A negative β„Ž, so plugging in a negative β„Ž, will shift the graph left, and if we would plug in a positive β„Ž, it would shift the graph right. π‘˜ moves the graph up and down, so if we plug in a positive π‘˜, it will shift up, and if we plug in a negative π‘˜, it will shift down, so β„Ž moves it horizontally and π‘˜ moves it vertically.

So first, let’s begin with π‘Ž. A dilation is what makes something bigger or smaller. And like we said, our graph could become thinner or wider; it’s the same thing. So our π‘Ž will be one-fourth.

Next, there’s a shift of four to the right, so we begin with π‘₯ minus and then we plug in β„Ž. So if we’re going four to the right, we’re plugging in a positive four. So even though it says π‘₯ minus four, we’re still going to the right.

And then lastly, we are shifting four up, so we plug in positive four for π‘˜. So this is in vertex form. Let’s go ahead and put it in standard form. So we need to evaluate this, in other words simplify.

So π‘₯ minus four squared means we need to multiply it to itself twice. So we FOIL. π‘₯ times π‘₯ is π‘₯ squared, π‘₯ times negative four is negative four π‘₯, and then we have negative four times π‘₯, so it’s another negative four π‘₯; that’s how we get our negative eight π‘₯.

And then lastly, we take negative four times negative four, which is 16. And now we need to take one-fourth times that polynomial, so we distribute the one-fourth and then we also bring down the four.

So our last step would be to combine the four and the four by adding. So the function 𝑓 is equal to π‘₯ squared divided by four minus two π‘₯ plus eight.

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