Question

Graph each function.

$f\left(x\right)=-\sqrt{64-16x^2}$

Short answer

The graph of the function is half an ellipse.

Step-by-step solution

Step 1. Given information.

WE have given function is $y=-\sqrt{64-16x^2}$.

Now, find an equation of conic, we have

$$\begin{gathered} y^2=64-16x^2 \\ {y}^2+16x^2=64 \\ \frac{y^2}{64}+\frac{x^2}{4}=1 \\ \frac{x^2}{2^2}+\frac{y^2}{8^2}=1 \end{gathered}$$

Step 2. Graph of the function.  

We have $b=2,a=8$in $b^2=a^2-c^2$

$$\begin{gathered} c^2=8^2-2^2 \\ {c}^2=60 \\ c=\sqrt{60} \\ c=2\sqrt{15} \end{gathered}$$

The foci are at $(0,\pm 2\sqrt{15})$.

The vertices of the major axis of the ellipse are at $(0,8)$and $(0,-8)$.

The vertices of the minor axis of the ellipse are at $(2,0)$and $(-2,0)$.

The graph of the function is