To solve this question, we need to analyze the given graph of the ellipse and identify its properties. Let's walk through the steps to find the standard form of the equation and determine the location of its foci.

1. Determine the Center: From the graph, it appears that the center of the ellipse is at the origin, $(0, 0)$.

2. Identify the Lengths of the Axes:

   • The semi-major axis appears to run horizontally (along the x-axis).
   • The graph shows that the semi-major axis has a length of about 5 (from -5 to 5 on the x-axis).
   • The semi-minor axis has a length of about 3 (from -3 to 3 on the y-axis).

3. Formulate the Equation: For an ellipse centered at the origin $(0, 0)$ with a horizontal major axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a$ is the semi-major axis and $b$ is the semi-minor axis.

   Since $a = 5$ and $b = 3$: $\frac{x^2}{5^2} + \frac{y^2}{3^2} = 1$ Simplifying, we get: $\frac{x^2}{25} + \frac{y^2}{9} = 1$

4. Find the Location of the Foci: The foci for an ellipse are located at a distance of $c$ from the center along the major axis, where $c$ is calculated using the formula: $c = \sqrt{a^2 - b^2}$ Substituting $a = 5$ and $b = 3$: $c = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ Thus, the foci are located at $( \pm 4, 0 )$ on the x-axis.

Final Answer:

• Equation of the ellipse: $\frac{x^2}{25} + \frac{y^2}{9} = 1$
• Location of the foci: $(4, 0)$ and $(-4, 0)$

Would you like more details on any part of this solution?

Additional Questions

1. How do you determine whether an ellipse's major axis is horizontal or vertical?
2. What is the formula for finding the foci of an ellipse centered at a point other than the origin?
3. How would the equation change if the ellipse were centered at $(h, k)$?
4. What steps are necessary to graph an ellipse given its standard equation?
5. How would the foci be affected if the semi-minor and semi-major axes were equal?

Tip: Remember, the values of $a$ and $b$ represent the semi-axes, so $a$ should always be greater than $b$ for ellipses with a horizontal or vertical major axis.