Chapter 9.1, Problem 64E

To determine

To find the Vertex, Focus and Directrix of a parabola.

Answer to Problem 64E

The values of $\text{vertex}=\left(-5,1\right),\text{focus}=\left(-\frac{21}{4},1\right)\text{ and directrix }\left(x=-\frac{19}{4}\right)$

Explanation of Solution

Given information:

The equation is

  ${\left(y-1\right)}^2=-\left(x+5\right)$

Calculation:

The Vertex, Focus and Directrix of a parabola can be obtained as

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

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

Comparing the given equation to the standard form of the equation it can observed that the vertex is

  $h=-5\text{, }k=1\text{ and }p=-\frac{1}{4}$

Add $p\text{ to }h$ to find the focus

  $\begin{array}{l}\text{focus = }\left(h+p,k\right)\\ \left(h+p,k\right)=\left(-5-\frac{1}{4},1\right)\left[\text{since }h=-5,k=1,p=-\frac{1}{4}\right]\\ \left(h+p,k\right)=\left(-\frac{21}{4},1\right)\end{array}$

Subtract $p\text{ to }h$ to find the directrix

  $\begin{array}{l}\text{Directrix, }\left(x=h-p\right)\\ \left(x=-5-\left(-\frac{1}{4}\right)\right)\\ \left(x=-5+\frac{1}{4}\right)\\ x=-\frac{19}{4}\end{array}$

Because p is negative, the parabola opens negative side of x-axis and the vertex of the parabola will be at $\left(-5,1\right)$ and the focus is at $\left(-\frac{21}{4},1\right)$ .

So, the graph is obtained as:

  

Hence,

The values of $\text{vertex}=\left(-5,1\right),\text{focus}=\left(-\frac{21}{4},1\right)\text{ and directrix }\left(x=-\frac{19}{4}\right)$

To verify the above graph, use graphing calculator to graph the given equation follow the steps given below:

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression ${\left(y-1\right)}^2=-\left(x+5\right)$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

Hence, the graph is verified.