The graph of the function 𝑓 is produced from that of 𝑔 of 𝑥 equals 𝑥 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. 𝑎, ℎ, and 𝑘 represent different things that we could do to our graph to change it or move it.
𝑎 represents the scale factor. It decides how thin or wide the graph will be. ℎ will shift our graph left and right. A negative ℎ, so plugging in a negative ℎ, will shift the graph left, and if we would plug in a positive ℎ, it would shift the graph right. 𝑘 moves the graph up and down, so if we plug in a positive 𝑘, it will shift up, and if we plug in a negative 𝑘, it will shift down, so ℎ moves it horizontally and 𝑘 moves it vertically.
So first, let’s begin with 𝑎. 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 𝑎 will be one-fourth.
Next, there’s a shift of four to the right, so we begin with 𝑥 minus and then we plug in ℎ. So if we’re going four to the right, we’re plugging in a positive four. So even though it says 𝑥 minus four, we’re still going to the right.
And then lastly, we are shifting four up, so we plug in positive four for 𝑘. 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 𝑥 minus four squared means we need to multiply it to itself twice. So we FOIL. 𝑥 times 𝑥 is 𝑥 squared, 𝑥 times negative four is negative four 𝑥, and then we have negative four times 𝑥, so it’s another negative four 𝑥; that’s how we get our negative eight 𝑥.
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 𝑥 squared divided by four minus two 𝑥 plus eight.