https://file.aiquickdraw.com/mathbot/file/1728571591174idyl4ab9.jpg

The image shows a hyperbola with its asymptotes labeled as \( y = 2x \) and \( y = -2x \). The graph is centered at the origin \( (0, 0) \), and the asymptotes indicate that the slopes are symmetrical but with opposite signs.

### Step-by-step process to find the equation of the hyperbola:
1. **Identify the general form**: For hyperbolas centered at the origin, the equation is generally either:
   \[
   \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad \text{(horizontal transverse axis)}
   \]
   or
   \[
   \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \quad \text{(vertical transverse axis)}.
   \]
   From the graph, we can see the hyperbola opens vertically (upward and downward), so we use the second form:
   \[
   \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1.
   \]

2. **Find the slopes of the asymptotes**: The equations of the asymptotes are given as \( y = 2x \) and \( y = -2x \). The slopes of these lines are \( \pm 2 \). For a hyperbola, the slopes of the asymptotes are given by \( \pm \frac{b}{a} \). So, we set:
   \[
   \frac{b}{a} = 2.
   \]
   This gives \( b = 2a \).

3. **Determine the values of \( a^2 \) and \( b^2 \)**: The hyperbola seems to extend horizontally by about 3 units on the graph (half the distance between the two vertices on the x-axis), so we estimate \( a = 3 \). Since \( b = 2a \), we have \( b = 6 \).

4. **Write the equation**: Using \( a = 3 \) and \( b = 6 \), the equation of the hyperbola becomes:
   \[
   \frac{y^2}{6^2} - \frac{x^2}{3^2} = 1,
   \]
   or simplified:
   \[
   \frac{y^2}{36} - \frac{x^2}{9} = 1.
   \]

Thus, the equation for the hyperbola is:
\[
\boxed{\frac{y^2}{36} - \frac{x^2}{9}} = 1.
\]

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Would you like more details on any part of the solution, or do you have any other questions?

Here are some related questions:
1. How do you derive the equation of a hyperbola with different center points?
2. What is the relationship between the asymptotes and the hyperbola's axes?
3. How can you graph a hyperbola if you are only given its equation?
4. How do you find the foci of a hyperbola from its equation?
5. What happens if a hyperbola's transverse axis is horizontal instead of vertical?

**Tip**: When identifying the equation of a hyperbola from a graph, always check the orientation (horizontal or vertical) and the asymptotes for clues about the equation's form.

Write an equation for the hyperbola shown in the graph, with asymptotes y = 2x and y = -2x.

Learn how to derive the equation of a hyperbola with asymptotes y = 2x and y = -2x. This problem involves understanding hyperbolic equations, asymptotes, and the relationship between the slopes and axes of the hyperbola. Suitable for Grades 11-12.

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