The given graph is an ellipse, and we need to identify the correct equation that represents it.

The standard form of the equation of an ellipse is:

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

where:

• $a$ is the distance from the center to the vertices along the x-axis (horizontal axis),
• $b$ is the distance from the center to the vertices along the y-axis (vertical axis).

From the graph:

• The ellipse stretches from $x = -6$ to $x = 6$, meaning the semi-major axis along the x-axis is 6. Therefore, $a^2 = 36$.
• The ellipse stretches from $y = -4$ to $y = 4$, meaning the semi-minor axis along the y-axis is 4. Therefore, $b^2 = 16$.

Thus, the equation of this ellipse is:

$\frac{x^2}{36} + \frac{y^2}{16} = 1$

Answer:

The correct option is:

$\boxed{\frac{x^2}{36} + \frac{y^2}{16} = 1}$

Would you like more details on the solution or any further explanation?

Here are 5 related questions to explore:

1. What is the general form of a hyperbola equation?
2. How do you determine whether an ellipse is horizontal or vertical?
3. What is the relationship between the focal points and the shape of an ellipse?
4. How do you calculate the eccentricity of an ellipse?
5. What is the difference between the equation of a circle and an ellipse?

Tip: The larger denominator in the equation of an ellipse corresponds to the axis that is longer!