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.