Question

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

$x^2-4x=2y$

Short answer

The vertex is $\left(2,-2\right)$, focus is $\left(2,\frac{-3}{2}\right)$and the directrix is $y=\frac{-5}{2}$.

The graph of the equation shown below .

Step-by-step solution

Step 1. Given information .

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

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 .

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

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

Further simplify .

Since the value of$a=\frac{1}{2}$the vertex is$\left(h,k\right)=\left(2,-2\right)$, focus is$\left(h,k+a\right)=\left(2,\frac{-3}{2}\right)$and the directrix is$y=k-a=\frac{-5}{2}$.

Step 3.  Plot the graph .

The graph of the equation $x^2-4x=2y$using graphing utility is shown below .

Here V is the vertex , F is focus and the line$y=\frac{-5}{2}$represents the directrix of the parabola .