Question

In the following exercises, graph each equation by translation$\frac{{\left(x-5\right)}^2}{9}+\frac{{\left(y+4\right)}^2}{25}=1$

Short answer

The graph of the ellipse $\frac{{\left(x-5\right)}^2}{9}+\frac{{\left(y+4\right)}^2}{25}=1$

Step-by-step solution

Step 1, Given the information 

The given equation of the ellipse is$\frac{{\left(x-5\right)}^2}{9}+\frac{{\left(y+4\right)}^2}{25}=1$

Step 2. Draw a graph 

This ellipse will have the same size and shape as $\frac{x^2}{9}+\frac{y^2}{25}=1$

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

Since $9<25$The major axis is vertical,

$$\begin{gathered} a^2=9 \\ a=\pm 3 \end{gathered}$$and $$\begin{gathered} b^2=25 \\ b=\pm 5 \end{gathered}$$

The vertices are $\left(0,5\right),(0,-5)$and The endpoints of the minor axis are $\left(3,0\right),\left(-3,0\right)$

Step 3. Sketch an original  graph

The Centre of the ellipse is $\left(5,-4\right)$

We translate the graph $\frac{x^2}{9}+\frac{y^2}{25}=1$

5 units to the right and then down 4 units.

Verify that the center is $\left(5,-4\right)$

Step 4. Conclusion

Hence, the new ellipse is the ellipse whose equation is

$\frac{{\left(x-5\right)}^2}{9}+\frac{{\left(y+4\right)}^2}{25}=1$