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.