Question

Write an equation for the parabola with the given vertex that passes through the given point.

Vertex $\left(0,8\right)$

Point $\left(4,0\right)$

Short answer

The equation of the parabola is$y=-\frac{x^2}{2}+8$.

Step-by-step solution

Step 1. Given Information.

Given, a parabola has the vertex $\left(0,8\right)$ and passes through the point $\left(4,0\right)$. The equation for the parabola is to be determined.

Step 2. Calculation.

The equation of a parabola with vertex $\left(h,k\right)$is $y=a{\left(x-h\right)}^2+k$

Here h=0,k=8 and since the parabola passes through the point $\left(4,0\right)$, x=4,y=0

Plugging the values in the equation and solving for a:

$\begin{array}{l}y=a{\left(x-h\right)}^2+k\\ 0=a{\left(4-0\right)}^2+8\\ a{\left(4\right)}^2=0-8\\ a\left(16\right)=-8\\ 16a=-8\\ a=-\frac{8}{16}\\ a=-\frac{1}{2}\end{array}$

Plugging back the value of a, h and k in the equation:

$\begin{array}{l}y=-\frac{1}{2}{\left(x-0\right)}^2+8\\ y=-\frac{x^2}{2}+8\end{array}$

Step 3. Conclusion.

Hence, the equation of the parabola is $y=-\frac{x^2}{2}+8$.