Chapter 9.1, Problem 76E

To determine

To find the standard equation of a parabola.

Answer to Problem 76E

The standard equation of a parabola is ${\left(x-3\right)}^2=\frac{12}{3}\left(y+3\right)$

Explanation of Solution

Given information:

From the graph given in the question we can observed that the vertex $\left(3,-3\right)$ and the value of $x=0\text{ and }y=0$

Calculation:

The standard equation of a parabola can be obtained as

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

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

Since the vertex is at $\left(3,-3\right)$ , so $\left(h,k\right)=\left(3,-3\right)$ .The point $\left(0,0\right)$ is on the parabola

Put the values $h=3,k=-3,x=0\text{ and }y=0$ in above equation

  $\begin{array}{l}{\left(0-3\right)}^2=4p\left(0-\left(-3\right)\right)\\ {\left(-3\right)}^2=4p\left(3\right)\\ 9=12p\\ p=\frac{12}{9}\\ p=\frac{4}{3}\left[\text{divide numerator and denominator by 3}\right]\end{array}$

Put the value of $p=\frac{4}{3}$ in the standard form of the equation of parabola

  $\begin{array}{l}{\left(x-h\right)}^2=4p\left(y-k\right)\\ {\left(x-3\right)}^2=4\left(\frac{4}{3}\right)\left(y-\left(-3\right)\right)\left[\text{since the vertex }\left(h,k\right)=\left(3,-3\right)\right]\\ {\left(x-3\right)}^2=\frac{12}{3}\left(y+3\right)\end{array}$

Hence,

The standard equation of a parabola is ${\left(x-3\right)}^2=\frac{12}{3}\left(y+3\right)$