Chapter 9.1, Problem 63E

To determine

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

Answer to Problem 63E

The values of $\text{vertex}=\left(\frac{3}{2},-7\right),\text{focus}=\left(\frac{5}{2},-7\right)\text{ and directrix }\left(x=\frac{1}{2}\right)$

Explanation of Solution

Given information:

The equation is

  ${\left(y+7\right)}^2=4\left(x-\frac{3}{2}\right)$

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

  $h=\frac{3}{2}\text{, }k=-7\text{ and }p=1$

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

  $\begin{array}{l}\text{focus = }\left(h+p,k\right)\\ \left(h+p,k\right)=\left(\frac{3}{2}+1,-7\right)\left[\text{since }h=\frac{3}{2},k=-7,p=1\right]\\ \left(h+p,k\right)=\left(\frac{5}{2},-7\right)\end{array}$

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

  $\begin{array}{l}\text{Directrix, }\left(x=h-p\right)\\ \left(x=\frac{3}{2}-1\right)\\ x=\frac{1}{2}\end{array}$

Because p is positive, the parabola opens positive side of x-axis and the vertex of the parabola will be at $\left(\frac{3}{2},-7\right)$ and the focus is at $\left(\frac{5}{2},-7\right)$ .

So, the graph is obtained as:

  

Hence,

The values of $\text{vertex}=\left(\frac{3}{2},-7\right),\text{focus}=\left(\frac{5}{2},-7\right)\text{ and directrix }\left(x=\frac{1}{2}\right)$

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 ${\left(y+7\right)}^2=4\left(x-\frac{3}{2}\right)$ 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.