The graph you've provided shows an ellipse. To find its equation, we use the general formula for an ellipse centered at the origin:

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

where:

• $a$ is the semi-major axis (the longest radius),
• $b$ is the semi-minor axis (the shortest radius).

Looking at the graph:

• The major axis (horizontal) has a length of 10, so the semi-major axis $a = 5$.
• The minor axis (vertical) has a length of 8, so the semi-minor axis $b = 4$.

Therefore, the equation of the ellipse is:

$\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1$

Simplifying:

$\frac{x^2}{25} + \frac{y^2}{16} = 1$

This is the equation of the ellipse shown in the graph.

Would you like further clarification on any step, or have additional questions? Here's a set of related questions:

1. How can we identify the center of the ellipse from the graph?
2. How would the equation change if the ellipse was shifted from the origin?
3. What happens to the shape of the ellipse if $a$ is smaller than $b$?
4. How can you graph the ellipse based on its equation?
5. What is the relationship between the foci and the equation of the ellipse?

Tip: For ellipses, the larger denominator corresponds to the axis of the ellipse that's longest (semi-major axis).