Question

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

Vertex $\left(-1,4\right)$

Point $\left(-2,2\right)$

Short answer

The equation of the parabola is$y=-2{\left(x+1\right)}^2+4$.

Step-by-step solution

Step 1. Given Information.

A given parabola has the vertex $\left(-1,4\right)$ and passes through the point $\left(-2,2\right)$. 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=-1,k=4 and since the parabola passes through the point $\left(-2,2\right)$, x=-2,y=2

Plugging the values in the equation and solving for a:

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

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

$\begin{array}{l}y=-2{\left(x-\left(-1\right)\right)}^2+4\\ y=-2{\left(x+1\right)}^2+4\end{array}$

Step 3. Conclusion.

Hence, the equation of the parabola is $y=-2{\left(x+1\right)}^2+4$.