### Home > A2C > Chapter 8 > Lesson 8.1.1 > Problem8-7

8-7.

Find the equation of the parabola that passes through the points $\left(0, 0\right)$, $\left(3, 9\right)$, and $\left(6, 0\right)$.

There are three ways to approach this problem.

Use the $x$-intercepts.

Notice that $\left(0, 0\right)$ and $\left(6, 0\right)$ are $x$-intercepts.
What are the factors?

The equation is $y = ax\left(x − 6\right)$.
Substitute $\left(3, 9\right)$ into the equation and solve for a.

Use the vertex.

Notice that since $3$ is midway between $0$ and $6$, $\left(3, 9\right)$ is the vertex.
Write the equation in vertex form.

The equation is
$y = a\left(x − 3\right)² + 9$.
Substitute one of the
$x$-intercepts into the equation and solve for $a$.

Standard form.

Substitute the three points into the equation
$y = ax² + bx + c$ to make a system of equations in three variables.

Since you know what $c$ is you can substitute it into all of the equations and solve a system of two equations in two variables.