Question

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

Vertex $\left(3,2\right)$

Point $\left(5,6\right)$

Short answer

The equation of the parabola is$y={\left(x-3\right)}^2+2$.

Step-by-step solution

Step 1. Given Information.

A given parabola has the vertex $\left(3,2\right)$ and passes through the point $\left(5,6\right)$. 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=3,k=2 and since the parabola passes through the point $\left(5,6\right)$, x=5,y=6

Plugging the values in the equation and solving for a:

$\begin{array}{l}y=a{\left(x-h\right)}^2+k\\ 6=a{\left(5-3\right)}^2+2\\ a{\left(5-3\right)}^2=6-2\\ a{\left(2\right)}^2=4\\ 4a=4\\ a=1\end{array}$

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

$\begin{array}{l}y=1{\left(x-3\right)}^2+2\\ y={\left(x-3\right)}^2+2\end{array}$

Step 3. Conclusion.

Hence, the equation of the parabola is $y={\left(x-3\right)}^2+2$.