EXPLANATION:
Given;
We are given the following equation of an ellipse;
[tex]9x^2+36x+4y^2-8y=-4[/tex]
Required;
We are required to graph the ellipse and state its center and vertices.
Step-by-step solution;
[tex]\begin{gathered} Re-write\text{ }in\text{ }standard\text{ }form: \\ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \end{gathered}[/tex]
We now have;
[tex]\begin{gathered} Our\text{ }ellipse\text{ }is\text{ }in\text{ }the\text{ }form: \\ \frac{(x-(-2)^2}{4}+\frac{(y-1)^2}{9}=1 \\ Hence: \\ \\ \frac{(x+2)^2}{2^2}+\frac{(y-1)^2}{3^2}=1 \end{gathered}[/tex]
With the center given as (h, k), we now have
[tex]Center=(-2,1)[/tex]
Also,
[tex]a=2,b=3[/tex]
Using a graphing tool, the graph would now appear as follows;
The vertices as shown in the graph are found at the point;
[tex]\begin{gathered} (h,k+b) \\ (h,k-b) \\ Hence: \\ (-2,1+3),(-2,1-3) \\ \\ Vertices=(-2,4),(-2,-2) \end{gathered}[/tex]
Therefore,
ANSWER:
[tex]\begin{gathered} Center=(-2,1) \\ \\ Vertices)=(-2,4),(-2-2) \end{gathered}[/tex]
