Question

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

$y^2=8x$

Short answer

The vertex is $\left(0,0\right)$focus is $\left(2,0\right)$and directrix is $x=-2$. The graph of the given equation is shown below .

Step-by-step solution

Step 1. Given information .

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

Step 2. Analyze the equation .

The standard form of parabola is $y^2=4ax$.

Compare the equation with standard form.

$$\begin{gathered} y^2=4ax \\ {y}^2=8x \\ 4a=8 \\ a=2 \end{gathered}$$

Further simplify .

Since $a=2$the vertex is $\left(0,0\right)$,the focus is $\left(a,0\right)=\left(2,0\right)$and the directrix is$x=-a,x=-2$

Step[ 3. Plot the graph .

The graph of the equation $y^2=8x$using graphing utility is shown below .

Here V is the vertex , F is the focus and line $x=-2$represents the directrix of the parabola .