Chapter 9.1, Problem 75E

To determine

To find the standard equation of a parabola.

Answer to Problem 75E

The standard equation of a parabola is ${\left(y-0\right)}^2=4\left(x+4\right)$

Explanation of Solution

Given information:

From the graph given in the question we can observed that the vertex $\left(-4,0\right)$ and the value of $x=0\text{ and }y=4$

Calculation:

The standard equation of a parabola can be obtained as

The standard form of the equation of a parabola with Horizontal axis and vertex at $\left(h,k\right)$ is

  ${\left(y-k\right)}^2=4p\left(x-h\right)$

Since the vertex is at $\left(-4,0\right)$ , so $\left(h,k\right)=\left(-4,0\right)$ .The point $\left(0,4\right)$ is on the parabola

Put the values $h=-4,k=0,x=0\text{ and }y=4$ in above equation

  $\begin{array}{l}{\left(4-0\right)}^2=4p\left(0-\left(-4\right)\right)\\ {\left(4\right)}^2=4p\left(0+4\right)\\ {\left(4\right)}^2=4p\left(4\right)\\ 16=16p\\ p=1\left[\text{divide both side by 16}\right]\end{array}$

Put the value of $p=1$ in the standard form of the equation of parabola

  $\begin{array}{l}{\left(y-k\right)}^2=4p\left(x-h\right)\\ {\left(y-0\right)}^2=4\left(1\right)\left(x-\left(-4\right)\right)\left[\text{since the vertex }\left(h,k\right)=\left(-4,0\right)\right]\\ {\left(y-0\right)}^2=4\left(x+4\right)\end{array}$

Hence,

The standard equation of a parabola is ${\left(y-0\right)}^2=4\left(x+4\right)$