Chapter 2.1, Problem 44E

To determine

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

Answer to Problem 44E

The standard form of equation for the parabola is $f\left(x\right)={\left(x+2\right)}^2-1$ .

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,

  $\left(h,k\right)=\left(-2,-1\right)$

Then, the equation for the graph is,

  $\begin{array}{c}f\left(x\right)=a{\left(x-\left(-2\right)\right)}^2-1\\ =a{\left(x+2\right)}^2-1\end{array}$ ..............(I)

The parabola passes through the point $\left(0,3\right)$ and it follows $f\left(x\right)=3$ . So,

  $\begin{array}{l}f\left(0\right)=a{\left(0+2\right)}^2-1\\ 3=a{\left(2\right)}^2-1\\ 3=4a-1\\ a=1\end{array}$

Substitute $1$ for $a$ in equation (I).

  $f\left(x\right)={\left(x+2\right)}^2-1$

Thus, the standard form of equation for the parabola is $f\left(x\right)={\left(x+2\right)}^2-1$ .