Question

In Problems 29–36, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.$x^2-y^2=4$

Short answer

$$\begin{gathered} \mathrm{Center}:(0,0) \\ \mathrm{Transverse}\mathrm{axis}:x-axis \\ \mathrm{Vertices}:(\pm 2,0) \\ \mathrm{Foci}:(\pm 2\sqrt{2},0) \\ \mathrm{Asymptotes}:y=\pm x \end{gathered}$$

Graph of the equation is:

Step-by-step solution

Step 1. Given information 

Equation of hyperbola, $x^2-y^2=4$

Writing the equation in standard form,$\frac{x^2}{4}-\frac{y^2}{4}=1$

Step 2. Finding the center, transverse axis, vertices, foci, and asymptotes of hyperbola

Comparing the given equation with standard form, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,b^2=c^2-a^2$

We get,

$$\begin{gathered} a=2,b=2, \\ c=\pm \sqrt{a^2+b^2} \\ \Rightarrow c=\pm \sqrt{2^2+2^2} \\ \Rightarrow c=\pm \sqrt{4+4} \\ \Rightarrow c=\pm \sqrt{8} \\ \Rightarrow c=\pm 2\sqrt{2} \end{gathered}$$

role="math" localid="1647143921159" $$\begin{gathered} \mathrm{Center}:(0,0) \\ \mathrm{Transverse}\mathrm{axis}:x-axis \\ \mathrm{Vertices}:(\pm \mathrm{a},0)=(\pm 2,0) \\ \mathrm{Foci}:(\pm \mathrm{c},0)=(\pm 2\sqrt{2},0) \\ \mathrm{Asymptotes}:y=\pm \frac{b}{a}x \\ \Rightarrow y=\pm \frac{2}{2}x \\ \Rightarrow y=\pm x \end{gathered}$$

Step 3. Graph of the equation

Graph of the equation is: