Chapter 9.1, Problem 71E

To determine

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

Answer to Problem 71E

The values of $\text{vertex}=\left(1,1\right),\text{focus}=\left(1,2\right)\text{ and directrix }\left(y=0\right)$

Explanation of Solution

Given information:

The equation is

  $y=\frac{1}{4}\left(x^2-2x+5\right)$

Calculation:

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

First convert the equation to standard form by completing the square.

  $\begin{array}{l}y=\frac{1}{4}\left(x^2-2x+5\right)\\ 4y=x^2-2x+5\\ x^2-2x=4y-5\\ x^2-2x+1=4y-5+1\left[\text{add 1 both sides}\right]\\ {\left(x-1\right)}^2=4y-4\\ {\left(x-1\right)}^2\text{=4}\left(y-1\right)\left[\text{standard form}\right]\end{array}$

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

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

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

  $h=1\text{, }k=1\text{ and }p=1$

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

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

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

  $\begin{array}{l}\text{Directrix, }\left(y=k-p\right)\\ \left(y=1-1\right)\\ \left(y=0\right)\\ y=0\end{array}$

Because p is positive, the parabola opens upwords and the vertex of the parabola will be at $\left(1,1\right)$ and the focus is at $\left(1,2\right)$ .

So, the graph is obtained as:

  

Hence,

The values of $\text{vertex}=\left(1,1\right),\text{focus}=\left(1,2\right)\text{ and directrix }\left(y=0\right)$

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 ${\left(x-1\right)}^2\text{=4}\left(y-1\right)$ 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.