Chapter 8.1, Problem 54E

Solution

To find: The equation for the parabola.

  ${\left(y+3\right)}^2=18\left(x-1\right)$

Given information:

To write an equation for the parabola.

  

Concept used:

Parabolas that open to the right or left are inverse relations of upward-or downward opening parabolas. The standard equation for the parabolas that open to the right or left is:

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

Explanation:

From the graph of the right opening parabola, it follows that:

The vertex $\left(h,k\right)$ is $\left(1,-3\right)$ .

The point $\left(x,y\right)$ on the parabola is $\left(5.5,0\right)$ .

Since the directrix is $5.5$ , the value of $p$ will be:

  $\begin{array}{l}x=h-p\\ \Rightarrow 5.5=1-p\\ \Rightarrow p=-4.5\end{array}$

The standard equation for the parabolas that open to the right or left is:

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

Substitute $k=-3,p=-4.5,$ and $h=1$ into the above equation:

  $\begin{array}{l}{\left(y-\left(-3\right)\right)}^2=4\left(4.5\right)\left(x-1\right)\\ \Rightarrow {\left(y+3\right)}^2=4\left(4.5\right)\left(x-1\right)\\ \Rightarrow {\left(y+3\right)}^2=18\left(x-1\right)\end{array}$