Question

Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

${\left(y+3\right)}^2=8\left(x-2\right)$

Short answer

The vertex is $\left(2,-3\right)$,focus is $\left(4,-3\right)$and the directrix is $x=0$.

The graph of the equation shown below.

Step-by-step solution

Step 1. Given information .

Consider the given equation${\left(y+3\right)}^2=8\left(x-2\right)$.

Step 2. Analyze the equation .

The standard form of parabola is ${\left(y-k\right)}^2=4a\left(x-h\right)$.

Compare the given equation with standard form of parabola.

$$\begin{gathered} {\left(y-k\right)}^2=4a\left(x-h\right) \\ {\left(y+3\right)}^2=8\left(x-2\right) \\ 4a=8 \\ a=2 \\ h=2,k=-3 \end{gathered}$$

Further simplify.

Since the value of $a=2$the vertex is$\left(h,k\right)=\left(2,-3\right)$,focus is$\left(h+a,k\right)=\left(4,-3\right)$and the directrix is$x=h-a=0$.

Step 3. Plot the graph .

The graph of the equation by using graphing utility just shown below.

Here V is the vertex , F is the focus and line $x=0$represents the line of the directrix .