Question

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

$y^2-2y=8x-1$

Short answer

The vertex is $\left(\frac{-1}{4},1\right)$,focus is $\left(\frac{7}{4},1\right)$and directrix is $x=\frac{-9}{4}$.

The graph of the equation shown below .

Step-by-step solution

Step 1. Given information .

Consider the given equation$y^2-2y=8x-1$.${\left(y-k\right)}^2=4a\left(x-h\right)$

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 .

role="math" localid="1647080511378" $$\begin{gathered} {\left(y-k\right)}^2=4a\left(x-h\right) \\ {\left(y-1\right)}^2=2\left(4x+1\right) \\ a=2 \\ h=\frac{-1}{4},k=1 \end{gathered}$$

Further simplify .

Since the value of$a=2$the vertex is$\left(h,k\right)=\left(\frac{-1}{4},1\right)$,focus is$\left(h+a,k\right)=\left(\frac{7}{4},1\right)$and directrix is$x=h-a=\frac{-9}{4}$.

Step 3. Plot the graph .

The graph of the parabola equation ${\left(y-1\right)}^2=2\left(4x+1\right)$using graphing utility is shown below .