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Video: Figuring Out the Rule of a Quadratic Function given Its Graph

Kathryn Kingham

What is the function whose graph is shown below?

02:28

Video Transcript

What is the function whose graph is shown below?

To solve this question, we’re going to use the vertex form for the equation of a parabola. Vertex form says: 𝑓 of π‘₯ equals a times π‘₯ minus β„Ž squared plus π‘˜, where β„Ž, π‘˜ is the minimum or maximum for that parabola.

We have a very clearly defined β„Ž and π‘˜. Our β„Ž-value would be the π‘₯-value of the minimum, and it said negative six. Our π‘˜-value where the height of the minimum is zero. Let’s take the β„Ž and the π‘˜ that we know and plug it into this vertex form.

We don’t know what a is, so we’ll just leave the a there. Then we’ll say π‘₯ minus our β„Ž, which is negative six. Now this is really important. The β„Ž-value in vertex form is being subtracted from π‘₯, and our β„Ž-value has a negative. So we need it to say π‘₯ minus negative six. And then we can square that value and add π‘˜, which is zero.

Now we wanna ask some questions about what our a-value does in vertex form. Here’s what we know. When a is greater than zero, our parabola will open up. When a is positive, our graph opens upward. When a is negative, when a is less than zero, our graph will open downward. Since our graph is opening upward, we would say that our a is a one value; it’s positive. Then we can simplify a little bit and change π‘₯ minus negative six to π‘₯ plus six. And we can leave off that zero since its no value since our π‘˜-value is zero.

The parabola graph to the left can be represented by 𝑓 of π‘₯ equals π‘₯ plus six squared.