Question

Analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility.

$\frac{{(x+4)}^2}{9}+\frac{{(y+2)}^2}{4}=1$

Short answer

The center of ellipse is at $(h,k)=(-4,-2)$, vertices are at $(-1,-2)and(-7,-2)$and foci are at $(-4\pm \sqrt{5},-2)$

The required graph is

Step-by-step solution

Step 1. Given Information 

The given equation is$\frac{{(x+4)}^2}{9}+\frac{{(y+2)}^2}{4}=1$

Step 2. Calculation   

Compare the given equation with the general form of equaton of ellipse,

$(h,k)=(-4,-2)$

Also, we have, $a^2=9,b^2=4$

The large number 9 is in the denominator of the term containing $x^2$. So, major axis is parallel to x-axis and major axis is $y=-2$

The vertices are at

$(h\pm a,k)=(-4\pm 3,-2)i.e.,(-1,-2)and(-7,-2)$

Since, we know that $c^2=a^2-b^2$, we get,

$$\begin{gathered} c^2=a^2-b^2 \\ {c}^2=9-4 \\ {c}^2=5 \end{gathered}$$

Thus, foci are at$\left(-4\pm \sqrt{5},-2\right)$

Step 3. Graph

On plotting all the points, we get the required graph,