91Ó°ÊÓ

The given function is

$f\left(x\right)=-\sqrt{9+9x^2}$

To graph the function, first convert the function into standard form of hyperbolas.

$f\left(x\right)=\sqrt{9+9x^2}$

Let us assume $f\left(x\right)=y$

So, $y=-\sqrt{9+9x^2}$

Squaring both sides,

$y^2=9+9x^2$

Substract $9x^2$from both sides,

$y^2-9x^2=9$

Divide by $9$,

$\frac{y^2}{9}-\frac{x^2}{1}=1$

This is the standard form of the hyperbolas.

Equation of a Hyperbola: Transverse axis along the $y$-axis

An equation of the hyperbola with center at $\left(0,0\right)$, foci at $\left(0,-c\right)$and $\left(0,c\right)$, and vertices at $\left(0,-a\right)$and $\left(0,a\right)$is

$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$; where $b^2=c^2-a^2$

The transverse axis is the$y$-axis.

Here, the equation of the hyperbola is

$\frac{y^2}{9}-x^2=1$

So, $a^2=9$

$a=3$

$b=1$

Since role="math" $c^2=a^2+b^2$,

$c^2=9+1$

role="math" $c^2=10$

$c=\sqrt{10}$

Thus, foci are $\left(0,-c\right)≡\left(0,-\sqrt{10}\right)$and $\left(0,c\right)≡\left(0,\sqrt{10}\right)$

The vertices are role="math" localid="1647165514747" $\left(0,-a\right)≡\left(0,-3\right)$and $\left(0,a\right)≡\left(0,3\right)$

The transverse axis is the $y$-axis.

The graph is