Calculus I - Common Graphs

Section 1.10 : Common Graphs

17. Without using a graphing calculator sketch the graph of $\displaystyle \frac{{{x^2}}}{{36}} - \frac{{{y^2}}}{{49}} = 1$.

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This is a hyperbola in standard form with the minus sign in front of the $y$ term and so will open right and left. The center of the hyperbola is at $\left( {0,0} \right)$, the two vertices are at $\left( { - 6,0} \right)$ and $\left( {6,0} \right)$, and the slope of the two asymptotes are $ \pm \frac{7}{6}$.

Here is a quick sketch of the hyperbola.

$The graph of the hyperbola $\frac{{{x^2}}}{{36}} - \frac{{{y^2}}}{{49}} = 1$, drawn in solid red, along with its two asymptotes, drawn as dashed blue lines. The hyperbola is centered at the origin and opens left and right with vertices at (-6,0) and (6,0). The asymptotes pass through the origin with slopes $\frac{7}{6}$ and $ - \frac{7}{6}$ and each branch of the hyperbola gets closer to them as it moves away from its vertex.$