https://r2.erweima.ai/mathbot/image/1725824825582grpo6wgy.jpg

The problem is asking for the equation that corresponds to the given graph, which is an ellipse.

### Analysis of the Graph

The graph shows an ellipse centered at the origin \((0, 0)\) with the following properties:

- **Major Axis**: Along the \(x\)-axis.
- **Minor Axis**: Along the \(y\)-axis.
- The lengths of the semi-axes can be determined by the points where the ellipse intersects the axes:
  - The ellipse intersects the \(x\)-axis at \(x = 4\) and \(x = -4\), so the semi-major axis (horizontal axis) length is 4.
  - The ellipse intersects the \(y\)-axis at \(y = 2\) and \(y = -2\), so the semi-minor axis (vertical axis) length is 2.

### Standard Form of an Ellipse

The standard form of the equation of an ellipse centered at the origin with its major axis along the \(x\)-axis is:

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

where:
- \(a\) is the semi-major axis (horizontal distance from the center to the ellipse along the \(x\)-axis).
- \(b\) is the semi-minor axis (vertical distance from the center to the ellipse along the \(y\)-axis).

### Identifying the Values of \(a\) and \(b\)

Given:
- \(a = 4\) (since the semi-major axis length is 4)
- \(b = 2\) (since the semi-minor axis length is 2)

Substitute these values into the equation:

\[
\frac{x^2}{4^2} + \frac{y^2}{2^2} = 1
\]

Simplify the expression:

\[
\frac{x^2}{16} + \frac{y^2}{4} = 1
\]

### Conclusion

The correct equation of the ellipse that matches the given graph is:

\[
\boxed{\frac{x^2}{16} + \frac{y^2}{4} = 1}
\]

This corresponds to the second option provided in the image.

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Would you like further details on this topic, or do you have any other questions?

### Additional Questions for Exploration:

1. What are the properties of an ellipse when the semi-major and semi-minor axes are equal?
2. How would the equation of the ellipse change if it were centered at a point other than the origin?
3. What is the general form of the equation for an ellipse with its axes not aligned with the coordinate axes?
4. How do you derive the foci of an ellipse from its standard equation?
5. Can you explain how to graph an ellipse given its equation in standard form?

### Tip:

When analyzing the equation of an ellipse, remember that the larger denominator under the squared terms represents the length of the semi-major axis.

Observe the graph. Which equation corresponds to it?

In this problem, you'll analyze a graph of an ellipse centered at the origin and identify the correct equation. Learn how to interpret the major and minor axes of the ellipse and apply the standard form of its equation.

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