Question

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

${\left(y-2\right)}^2=8\left(x+1\right)$

Short answer

The vertex is $\left(-1,2\right)$,focus is $\left(1,2\right)$and directrix is $x=-3$.

The graph of the equation shown below .

Step-by-step solution

Step 1. Given information .

Consider the given equation${\left(y-2\right)}^2=8\left(x+1\right)$.

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.

$$\begin{gathered} {\left(y-k\right)}^2=4a\left(x-h\right) \\ {\left(y-2\right)}^2=8\left(x+1\right) \end{gathered}$$

role="math" localid="1646822176449" $$\begin{gathered} 4a=8 \\ a=2 \\ h=-1,k=2 \end{gathered}$$

Since the value of $a=1$the vertex is $\left(h,k\right)=\left(-1,2\right)$and focus is role="math" localid="1646822293613" $\left(h+a,k\right)=\left(1,2\right)$and the directrix is$x=h-a=-1-2=-3$

Step 3. Plot the graph.

BY using graphic utility the graph of the given equation ${\left(y-2\right)}^2=8\left(x+1\right)$shown below .

Here V is the vertex ,F is the focus and line $x=-3$represents the directrix of the parabola .