Graphing an Ellipse Centered at (h, k)

First, we determine the position of the major axis. Because $9>4,$ the major axis is parallel to the y-axis. Therefore, the equation is in the form $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1,$ where $b^2=4$ and $a^2=9.$ It follows that:

• the center of the ellipse is $\left(h,k\right)=\left(\mathrm{-2},\text{5}\right)$
• the coordinates of the vertices are $(h,k\pm a)=(-2,5\pm \sqrt{9})=(-2,5\pm 3),$ or $\left(\mathrm{-2},\text{2}\right)$ and $\left(\mathrm{-2},\text{8}\right)$
• the coordinates of the co-vertices are $(h\pm b,k)=(-2\pm \sqrt{4},5)=(-2\pm 2,5),$ or and $\left(0,\text{5}\right)$
• the coordinates of the foci are $\left(h,k\pm c\right),$ where $c^2=a^2-b^2.$ Solving for $c,$ we have:

Therefore, the coordinates of the foci are $\left(\mathrm{-2},\text{5}-\sqrt{5}\right)$ and $\left(\mathrm{-2},\text{5+}\sqrt{5}\right).$

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.