Question

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

$x^2+8x=4y-8$

Short answer

The vertex is $\left(-4,-2\right)$,focus is $\left(-4,-1\right)$and directrix is $y=-3$.

The graph of the equation is shown below .

Step-by-step solution

Step 1.  Given information .

Consider the given equation$x^2+8x=4y-8$.

Step 2. Analyze the equation .

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

Compare the given equation with standard form .

role="math" localid="1647078278628" $$\begin{gathered} {\left(x-h\right)}^2=4a\left(y-k\right) \\ {\left(x+4\right)}^2=4\left(y+2\right) \\ 4a=4 \\ a=1 \\ h=-4,k=-2 \end{gathered}$$

Further simplify .

Since the value of $a=1$then the vertex is $\left(h,k\right)=\left(-4,-2\right)$and focus is$\left(h,k+a\right)=\left(-4,-1\right)$and directrix is$y=k-a=-3$.

Step 3. Plot the graph .

The graph of the parabola equation ${\left(x+4\right)}^2=4\left(y+2\right)$using graphing utility is shown below .