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The graph depicts an ellipse centered at \((-6, 0)\), and we need to write its equation in standard form. The standard form of the equation for an ellipse is:

\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]

Where:
- \((h, k)\) is the center of the ellipse,
- \(a\) is the distance from the center to the vertices along the x-axis,
- \(b\) is the distance from the center to the vertices along the y-axis.

### Steps to write the equation:

1. **Center of the ellipse**:  
   The center appears to be at \((-6, 0)\).

2. **Horizontal axis length (major axis)**:  
   The ellipse extends horizontally from \(-10\) to \(-2\) on the x-axis. The total distance is:
   \[
   10 - 2 = 8
   \]
   Therefore, \(a = 4\) because the horizontal distance from the center to the vertices is half of the total length.

3. **Vertical axis length (minor axis)**:  
   The ellipse extends vertically from \(0\) to about \(\pm 2\), meaning the total vertical distance is:
   \[
   2 - (-2) = 4
   \]
   Therefore, \(b = 2\) because the vertical distance from the center to the vertices is half of the total length.

### Equation of the ellipse:
Substituting the values \(h = -6\), \(k = 0\), \(a = 4\), and \(b = 2\) into the standard equation:

\[
\frac{(x + 6)^2}{4^2} + \frac{y^2}{2^2} = 1
\]

Simplifying:

\[
\frac{(x + 6)^2}{16} + \frac{y^2}{4} = 1
\]

This is the equation of the ellipse in standard form.

Would you like further details on how this equation was derived or any specific explanation? Here are a few related questions you can explore:

1. How do you determine the center of an ellipse from a graph?
2. What are the differences between the standard forms of an ellipse and a hyperbola?
3. How does the major axis of an ellipse affect its shape?
4. Why is it important to square the values of \(a\) and \(b\) in the equation?
5. How do you convert the equation of an ellipse from general form to standard form?

**Tip**: Always ensure that the larger value, between \(a\) and \(b\), represents the axis along which the ellipse is longer (major axis).

The graph shows an ellipse. Write its equation in standard form.

Discover how to derive the equation of an ellipse from a graph, with a clear example centered at (-6, 0). This tutorial explains each step to formulate the ellipse equation using the standard conic sections format. Suitable for high school students in Grades 10-12.

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