Graphing Hyperbolas Worksheet for 11th Grade | Lesson Planet - Free Printable

To solve the problem of graphing hyperbolas, we need to understand the standard forms of hyperbola equations and their properties. The given equations are in the form:

1. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
2. $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$

Step-by-Step Solution:

#### Problem 1: Graph the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$

1. Identify the type of hyperbola:
- The equation is in the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, which represents a hyperbola centered at the origin $(0, 0)$ with a horizontal transverse axis.

2. Determine $a$ and $b$:
- From $\frac{x^2}{9} - \frac{y^2}{4} = 1$, we have $a^2 = 9$ and $b^2 = 4$.
- Therefore, $a = 3$ and $b = 2$.

3. Find the vertices:
- The vertices are located at $(\pm a, 0)$. So, the vertices are at $(\pm 3, 0)$.

4. Find the foci:
- The distance from the center to each focus is given by $c$, where $c = \sqrt{a^2 + b^2}$.
- Here, $c = \sqrt{9 + 4} = \sqrt{13}$.
- The foci are at $(\pm c, 0)$, so the foci are at $(\pm \sqrt{13}, 0)$.

5. Find the asymptotes:
- The equations of the asymptotes for a hyperbola of this form are $y = \pm \frac{b}{a}x$.
- Here, the asymptotes are $y = \pm \frac{2}{3}x$.

6. Graph the hyperbola:
- Plot the center at $(0, 0)$.
- Mark the vertices at $(\pm 3, 0)$.
- Draw the asymptotes $y = \pm \frac{2}{3}x$.
- Sketch the hyperbola branches approaching the asymptotes but not crossing them.

#### Problem 2: Graph the hyperbola $\frac{y^2}{4} - \frac{x^2}{9} = 1$

1. Identify the type of hyperbola:
- The equation is in the form $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$, which represents a hyperbola centered at the origin $(0, 0)$ with a vertical transverse axis.

2. Determine $a$ and $b$:
- From $\frac{y^2}{4} - \frac{x^2}{9} = 1$, we have $b^2 = 4$ and $a^2 = 9$.
- Therefore, $a = 3$ and $b = 2$.

3. Find the vertices:
- The vertices are located at $(0, \pm b)$. So, the vertices are at $(0, \pm 2)$.

4. Find the foci:
- The distance from the center to each focus is given by $c$, where $c = \sqrt{a^2 + b^2}$.
- Here, $c = \sqrt{9 + 4} = \sqrt{13}$.
- The foci are at $(0, \pm c)$, so the foci are at $(0, \pm \sqrt{13})$.

5. Find the asymptotes:
- The equations of the asymptotes for a hyperbola of this form are $y = \pm \frac{b}{a}x$.
- Here, the asymptotes are $y = \pm \frac{2}{3}x$.

6. Graph the hyperbola:
- Plot the center at $(0, 0)$.
- Mark the vertices at $(0, \pm 2)$.
- Draw the asymptotes $y = \pm \frac{2}{3}x$.
- Sketch the hyperbola branches approaching the asymptotes but not crossing them.

#### Problems 3, 4, 5, and 6:
- These problems follow the same steps as Problems 1 and 2. You need to identify whether the hyperbola has a horizontal or vertical transverse axis, determine $a$ and $b$, find the vertices, foci, and asymptotes, and then graph the hyperbola.

Final Answer:

The solutions involve graphing the hyperbolas based on the steps outlined above. For each problem, you need to:
1. Identify the type of hyperbola (horizontal or vertical transverse axis).
2. Determine $a$ and $b$.
3. Find the vertices, foci, and asymptotes.
4. Sketch the hyperbola.

For Problem 1: $\boxed{\text{Graph the hyperbola } \frac{x^2}{9} - \frac{y^2}{4} = 1 \text{ with vertices at } (\pm 3, 0), \text{ foci at } (\pm \sqrt{13}, 0), \text{ and asymptotes } y = \pm \frac{2}{3}x.}$

For Problem 2: $\boxed{\text{Graph the hyperbola } \frac{y^2}{4} - \frac{x^2}{9} = 1 \text{ with vertices at } (0, \pm 2), \text{ foci at } (0, \pm \sqrt{13}), \text{ and asymptotes } y = \pm \frac{2}{3}x.}$

Repeat similar steps for Problems 3, 4, 5, and 6.