# What is the equation of the parabola with axis intercepts of x=2, x=-4, and y=4?

May 22, 2018

$y = - \frac{1}{2} {x}^{2} - x + 4$

#### Explanation:

The $x$ intercepts ($x = 2$ and $x = - 4$) are the zeros of the quadratic function, corresponding to linear factors $\left(x - 2\right)$ and $\left(x + 4\right)$.

So our equation can be written something like:

$y = a \left(x - 2\right) \left(x + 4\right)$

for some constant $a$ to be determined.

Putting $x = 0$ and $y = 4$, we have:

$\textcolor{b l u e}{4} = a \left(\textcolor{b l u e}{0} - 2\right) \left(\textcolor{b l u e}{0} + 4\right) = - 8 a$

Hence $a = - \frac{1}{2}$ and we have:

$y = - \frac{1}{2} \left(x - 2\right) \left(x + 4\right) = - \frac{1}{2} \left({x}^{2} + 2 x - 8\right) = - \frac{1}{2} {x}^{2} - x + 4$