College Algebra, Chapter 2, 2.3, Section 2.3, Problem 30

Graph the hyperbola $y^2 - 9x^2 = 1$ by solving for $y$ and graphing the two equations corresponding to the positive and negative squareroots.

$\begin{equation} \begin{aligned} y ^2 - 9x^2 &= 1 && \text{Model}\\ \\ y^2 &= 1 + 9x^2 && \text{Add } 9x^2 \\ \\ y &= \pm \sqrt{1 + 9x^2} && \text{take the square root} \end{aligned} \end{equation}$

Thus, the hyperbola is described by the graphs of two equations
$y = \sqrt{1+9x^2}$ and $y = -\sqrt{1+9x^2}$

The first equation represents the positive half of the hyperbola because $y \geq 0$ while the second represents the negative half. If we graph the first equation in the viewing recatngle $[-2,2]$ by $[-3,3]$ then we get...



The graph of the second equation is



Graphing these half portions together on the same viewing screen, we get the full hyperbola