Question

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

${\left(x-2\right)}^2=4\left(y-3\right)$

Short answer

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

The graph of the equation ${\left(x-2\right)}^2=4\left(y-3\right)$shown below .

Step-by-step solution

Step 1. Given information .

Consider the given equation${\left(x-2\right)}^2=4\left(y-3\right)$.

Step 2. Analyze the equation .

The standard form of parabola is ${\left(x-h\right)}^2=4\left(y-3\right)$.

Compare the given equation with standard form.

$$\begin{gathered} {\left(x-h\right)}^2=4a\left(y-k\right) \\ {\left(x-2\right)}^2=4\left(y-3\right) \\ 4a=4 \\ a=1 \\ h=2,k=3 \end{gathered}$$

Further simplify.

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

Step 3. Plot the graph .

By using the graphing utility the graph of the equation ${\left(x-2\right)}^2=4\left(y-3\right)$just shown below .

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