Chapter 9, Problem 2STP

To determine

To find: The correct option for the equation of the parabola that has the vertex $\left(-3,-23\right)$ and passes through the point $\left(1,9\right)$ .

Answer to Problem 2STP

The correct option is $\text{H}$ .

Explanation of Solution

Given information:

The given vertex is $\left(-3,-23\right)$ .

The parabola passes through the point $\left(1,9\right)$ .

Calculation:

The standard form of the parabola is.

  $y=a{\left(x-h\right)}^2+k$

Substitute the value $\left(h,k\right)=\left(-3,-23\right)$ in above equation.

  $\begin{array}{c}y=a{\left(x-\left(-3\right)\right)}^2+\left(-23\right)\\ =a{\left(x+3\right)}^2-23\end{array}$

The parabola passes through the point $\left(1,9\right)$ .

Substitute the value $\left(x,y\right)=\left(1,9\right)$ in above equation.

  $\begin{array}{c}9=a{\left(1+3\right)}^2-23\\ 9+23=a{\left(4\right)}^2\\ a=2\end{array}$

Substitute $2$ for $a$ in equation of parabola.

  $\begin{array}{c}y=2{\left(x+3\right)}^2-23\\ =2x^2+12x+18-23\\ =2x^2+12x-5\end{array}$

Therefore, the correct option is $\text{H}$ .