Question

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

${\left(y+1\right)}^2=-4\left(x-2\right)$

Short answer

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

Step-by-step solution

Step 1. Given information .

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

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.

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

Further simplify.

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

Step 3. Plot the graph .

The graph of the equation by using graphing utility just shown below.

Here V is the vertex , F is the focus and line $x=3$represents directrix of the parabola .