Calculus I - Common Graphs

Section 1.10 : Common Graphs

18. Without using a graphing calculator sketch the graph of $\displaystyle {\left( {y + 2} \right)^2} - \frac{{{{\left( {x + 4} \right)}^2}}}{{16}} = 1$.

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This is a hyperbola in standard form with the minus sign in front of the $x$ term and so will open up and down. The center of the hyperbola is at $\left( { - 4, - 2} \right)$, the two vertices are at $\left( { - 4, - 1} \right)$ and $\left( { - 4, - 3} \right)$, and the slope of the two asymptotes are $ \pm \frac{1}{4}$.

Here is a quick sketch of the hyperbola.

$The graph of the hyperbola ${\left( {y + 2} \right)^2} - \frac{{{{\left( {x + 4} \right)}^2}}}{{16}} = 1$, drawn in solid red, along with its two asymptotes, drawn as dashed blue lines. The hyperbola is centered at (-4,-2) and opens up and down with vertices at (-4,-1) and (-4,-3). The asymptotes cross at the center with slopes $\frac{1}{4}$ and $ - \frac{1}{4}$ and each branch of the hyperbola gets closer to them as it moves away from its vertex.$