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).