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

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 𝑥 equals a times 𝑥 minus ℎ squared plus 𝑘, where ℎ, 𝑘 is the minimum or maximum for that parabola.

We have a very clearly defined ℎ and 𝑘. Our ℎ-value would be the 𝑥-value of the minimum, and it said negative six. Our 𝑘-value where the height of the minimum is zero. Let’s take the ℎ and the 𝑘 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 𝑥 minus our ℎ, which is negative six. Now this is really important. The ℎ-value in vertex form is being subtracted from 𝑥, and our ℎ-value has a negative. So we need it to say 𝑥 minus negative six. And then we can square that value and add 𝑘, 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 𝑥 minus negative six to 𝑥 plus six. And we can leave off that zero since its no value since our 𝑘-value is zero.

The parabola graph to the left can be represented by 𝑓 of 𝑥 equals 𝑥 plus six squared.