Question

In the following exercises, graph each equation by translation

$\frac{\left(x+6^2\right)}{16}+\frac{{\left(y+5\right)}^2}{4}=1$

Short answer

The graph of the ellipse $\frac{{\left(x+6\right)}^2}{16}+\frac{{(y+5)}^2}{4}=1$is

Step-by-step solution

Step 1, Given the information  

The given equation of the ellipse is$\frac{\left(x+6^2\right)}{16}+\frac{{\left(y+5\right)}^2}{4}=1$

Step 2. Draw a graph  

This ellipse will have the same size and shape as $\frac{x^2}{16}+\frac{y^2}{4}=1$

Whose center is $\left(0,0\right)$Firstly we plot this graph,

Since$16>4$,The major axis is horizontal

$$\begin{gathered} a^2=16 \\ a=\pm 4 \end{gathered}$$and $$\begin{gathered} b^2=4 \\ b=\pm 2 \end{gathered}$$

The vertices are $\left(4,0\right)and(-4,0)$and The endpoints of the minor axis are $\left(0,2\right)and(0,-2)$

Step 3. Sketch an original  graph 

The Centre of the ellipse is $\left(-6,-5\right)$

We translate the graph $\frac{x^2}{16}+\frac{y^2}{4}=1$

6 units to the left and then down 5 units.

Verify that the center is $\left(-6,-5\right)$

Step 4. Conclusion 

Hence, the new ellipse is the ellipse whose equation is

$\frac{{\left(x+6\right)}^2}{16}+\frac{{(y+5)}^2}{4}=1$