# How to find quadratic function in vertex form from two points?

I'm starting to learn about quadratic formulas in math class. This question came up in a homework packet:

1. A WNBA player takes a three-point shot 22 feet away from the basket, The ball reaches its highest point, 16 feet above the floor, when it is 12 feet (horizontal distance) away from the player. As the ball approaches the basket, it is 13.6 feet above the floor when it is 18 feet away from the player.

a. Write a function in vertex form to model the shot of the basketball. (Let $x$ be the horizontal distance from the player and $y$ be the height of the basketball above the floor).

From today's lesson on quadratic formulas in the vertex form, the vertex form is $y=a(x-h)^2+k$, where $a$ is the growth factor, $h$ is the line of symmetry (x-coordinate of vertex), and $k$ is the y-coordinate of the vertex.

From the question, I can get that the vertex is (12,16), so $h$ is 12 and $k$ is 16. So that means the equation is $y=?(x-12)^2+16$. Another point is (18,13.6). I might be able to infer that another point is (22,0), but it wouldn't help much. $a$ is missing.

There is no more information given. My teacher said that many of us would most likely come back tomorrow begging her for answers, so maybe the answer is not purely mathematical, but requires thinking. How do I solve this (for $a$)?

Thanks.

Plug in $(x,y)=(18, 13.6)$ to get $13.6=a(18-12)^2+16$ and then solve for $a$. You can do this because you know the point $(18, 13.6)$ is on the parabola, so it must satisfy the equation.