Chapter 2.1, Problem 45E

To determine

To find: The equation for the parabola in the standard form.

Answer to Problem 45E

The standard form of equation for the parabola is $f\left(x\right)=-2{\left(x+2\right)}^2+2$ .

Explanation of Solution

Given:

The given graph for the parabola is shown in Figure 1

  

Figure 1

Calculation:

Consider the quadratic function is,

  $f\left(x\right)=a{\left(x-h\right)}^2+k$

The vertex of the above parabola for the standard form $f\left(x\right)=a{\left(x-h\right)}^2+k$ is,

  $\left(h,k\right)$

The vertex of the given graph is,

  $\left(h,k\right)=\left(-2,2\right)$

Then, the equation for the graph is,

  $\begin{array}{c}f\left(x\right)=a{\left(x-\left(-2\right)\right)}^2+2\\ =a{\left(x+2\right)}^2+2\end{array}$ ..............(I)

The parabola passes through the point $\left(-1,0\right)$ and it follows $f\left(-1\right)=0$ . So,

  $\begin{array}{l}f\left(-1\right)=a{\left(-1-\left(-2\right)\right)}^2+2\\ 0=a{\left(-1+2\right)}^2+2\\ a+2=0\\ a=-2\end{array}$

Substitute $-2$ for $a$ in equation (I).

  $f\left(x\right)=-2{\left(x+2\right)}^2+2$

Thus, the standard form of equation for the parabola is $f\left(x\right)=-2{\left(x+2\right)}^2+2$ .