To solve this problem, we need to identify the equation of the ellipse based on the graph.

Standard Equation of an Ellipse:

The equation of an ellipse centered at the origin $(0,0)$ is given by:

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

Where:

• $a$ is the semi-major axis (the distance from the center to the farthest point on the ellipse along the x-axis or y-axis).
• $b$ is the semi-minor axis (the distance from the center to the shortest point on the ellipse along the x-axis or y-axis).

Step-by-Step Solution:

1. Identify the Semi-Major and Semi-Minor Axes:

   • From the graph, the ellipse is wider along the x-axis. This means the major axis is along the x-axis, and the semi-major axis length is $a = 4$.
   • The minor axis is along the y-axis, and the semi-minor axis length is $b = 3$.

2. Write the Equation:

   • Using the values $a = 4$ and $b = 3$, the equation of the ellipse is:

$\frac{x^2}{4^2} + \frac{y^2}{3^2} = 1$

Which simplifies to:

$\frac{x^2}{16} + \frac{y^2}{9} = 1$

Answer:

The correct equation of the ellipse is:

$\frac{x^2}{16} + \frac{y^2}{9} = 1$

Thus, the first option highlighted is correct.

Would you like further explanation on ellipses or their properties?

Here are some related questions:

1. How can we differentiate between an ellipse and a circle graphically?
2. What happens if the semi-major axis is along the y-axis instead?
3. How do you calculate the foci of an ellipse?
4. Can an ellipse equation have a center other than the origin?
5. What changes in the equation if we rotate the ellipse?

Tip: For ellipses, always check which axis (x or y) is longer to identify the semi-major and semi-minor axes.