## Graph Hyperbolas

### Learning Outcomes

• Graph a hyperbola centered at the origin.
• Graph a hyperbola not centered at the origin.

When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form $\dfrac{{x}^{2}}{{a}^{2}}-\dfrac{{y}^{2}}{{b}^{2}}=1$ for horizontal hyperbolas and the standard form $\dfrac{{y}^{2}}{{a}^{2}}-\dfrac{{x}^{2}}{{b}^{2}}=1$ for vertical hyperbolas.

### How To: Given a standard form equation for a hyperbola centered at $\left(0,0\right)$, sketch the graph.

• Determine which of the standard forms applies to the given equation.
• Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the vertices, co-vertices, and foci; and the equations for the asymptotes.
• If the equation is in the form $\dfrac{{x}^{2}}{{a}^{2}}-\dfrac{{y}^{2}}{{b}^{2}}=1$, then
• the transverse axis is on the x-axis
• the coordinates of the vertices are $\left(\pm a,0\right)$
• the coordinates of the co-vertices are $\left(0,\pm b\right)$
• the coordinates of the foci are $\left(\pm c,0\right)$
• the equations of the asymptotes are $y=\pm \frac{b}{a}x$
• If the equation is in the form $\dfrac{{y}^{2}}{{a}^{2}}-\dfrac{{x}^{2}}{{b}^{2}}=1$, then
• the transverse axis is on the y-axis
• the coordinates of the vertices are $\left(0,\pm a\right)$
• the coordinates of the co-vertices are $\left(\pm b,0\right)$
• the coordinates of the foci are $\left(0,\pm c\right)$
• the equations of the asymptotes are $y=\pm \frac{a}{b}x$
• Solve for the coordinates of the foci using the equation $c=\pm \sqrt{{a}^{2}+{b}^{2}}$.
• Plot the vertices, co-vertices, foci, and asymptotes in the coordinate plane, and draw a smooth curve to form the hyperbola.

### Example: Graphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form

Graph the hyperbola given by the equation $\dfrac{{y}^{2}}{64}-\dfrac{{x}^{2}}{36}=1$. Identify and label the vertices, co-vertices, foci, and asymptotes.

### Try It

Graph the hyperbola given by the equation $\dfrac{{x}^{2}}{144}-\dfrac{{y}^{2}}{81}=1$. Identify and label the vertices, co-vertices, foci, and asymptotes.

## Graphing Hyperbolas Not Centered at the Origin

Graphing hyperbolas centered at a point $\left(h,k\right)$ other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms $\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}-\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$ for horizontal hyperbolas, and $\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}-\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}=1$ for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

### How To: Given a general form for a hyperbola centered at $\left(h,k\right)$, sketch the graph.

• Convert the general form to that standard form. Determine which of the standard forms applies to the given equation.
• Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the center, vertices, co-vertices, foci; and equations for the asymptotes.
• If the equation is in the form $\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}-\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$, then
• the transverse axis is parallel to the x-axis
• the center is $\left(h,k\right)$
• the coordinates of the vertices are $\left(h\pm a,k\right)$
• the coordinates of the co-vertices are $\left(h,k\pm b\right)$
• the coordinates of the foci are $\left(h\pm c,k\right)$
• the equations of the asymptotes are $y=\pm \frac{b}{a}\left(x-h\right)+k$
• If the equation is in the form $\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}-\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}=1$, then
• the transverse axis is parallel to the y-axis
• the center is $\left(h,k\right)$
• the coordinates of the vertices are $\left(h,k\pm a\right)$
• the coordinates of the co-vertices are $\left(h\pm b,k\right)$
• the coordinates of the foci are $\left(h,k\pm c\right)$
• the equations of the asymptotes are $y=\pm \frac{a}{b}\left(x-h\right)+k$
• Solve for the coordinates of the foci using the equation $c=\pm \sqrt{{a}^{2}+{b}^{2}}$.
• Plot the center, vertices, co-vertices, foci, and asymptotes in the coordinate plane and draw a smooth curve to form the hyperbola.

### Example: Graphing a Hyperbola Centered at (h, k) Given an Equation in General Form

Graph the hyperbola given by the equation $9{x}^{2}-4{y}^{2}-36x - 40y - 388=0$. Identify and label the center, vertices, co-vertices, foci, and asymptotes.

### Try It

Graph the hyperbola given by the standard form of an equation $\dfrac{{\left(y+4\right)}^{2}}{100}-\dfrac{{\left(x - 3\right)}^{2}}{64}=1$. Identify and label the center, vertices, co-vertices, foci, and asymptotes.

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