https://file.aiquickdraw.com/mathbot/file/1726517880540s18cf47j.jpg

The given equation of the ellipse is:

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

This is the standard form of the equation of an ellipse centered at the origin \((0, 0)\), where:

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

Here, \(a^2 = 25\) and \(b^2 = 16\), which gives us:
- \(a = \sqrt{25} = 5\) (the length of the semi-major axis along the x-axis)
- \(b = \sqrt{16} = 4\) (the length of the semi-minor axis along the y-axis)

### Steps to sketch the ellipse:
1. **Center**: The ellipse is centered at the origin \((0, 0)\).
2. **Semi-Major Axis**: The semi-major axis is along the x-axis with length \(5\). So, the ellipse stretches from \((-5, 0)\) to \((5, 0)\).
3. **Semi-Minor Axis**: The semi-minor axis is along the y-axis with length \(4\). So, the ellipse stretches from \((0, -4)\) to \((0, 4)\).
4. **Foci**: The foci of the ellipse are located at a distance \(c\) from the center, where \(c = \sqrt{a^2 - b^2}\). Here:
   \[
   c = \sqrt{25 - 16} = \sqrt{9} = 3
   \]
   Thus, the foci are located at \((\pm 3, 0)\) along the x-axis.

### Key Points for the Graph:
- The ellipse is wider along the x-axis because the semi-major axis is longer.
- The foci are positioned at \((3, 0)\) and \((-3, 0)\).

Would you like a detailed diagram, or do you have any questions about this process?

Here are some related questions for further exploration:
1. How do you find the eccentricity of an ellipse, and what does it represent?
2. What happens to the shape of the ellipse if \(a\) and \(b\) are equal?
3. Can you explain how the formula for the foci of an ellipse is derived?
4. How would the equation change if the ellipse were centered at a different point?
5. What if the semi-major axis were along the y-axis instead of the x-axis?

**Tip**: When graphing ellipses, always identify the lengths of the semi-major and semi-minor axes first, as this determines the overall shape and orientation.

Sketch the graph of an ellipse that has the following equation: (x^2 / 25) + (y^2 / 16) = 1. Include the foci.

Learn how to graph the ellipse with the equation (x^2 / 25) + (y^2 / 16) = 1, including how to calculate the foci using the Pythagorean theorem. The semi-major and semi-minor axes are discussed with a detailed solution suitable for Grades 10-12.

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