Question

the graph shows an ellipse. write its equation in standard form.

Explanation:

Step1: Identify the center of the ellipse

The center of the ellipse is at the point \((- 4,-7)\) as it is the mid - point of the ellipse.

Step2: Determine the semi - major axis \(a\) and semi - minor axis \(b\)

By counting the grid squares, the horizontal distance from the center to the edge of the ellipse (semi - minor axis \(b\)) is 2 units and the vertical distance from the center to the edge of the ellipse (semi - major axis \(a\)) is 3 units.

Step3: Write the standard form of the ellipse equation

The standard form of an ellipse with center \((h,k)\) is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\). Here \(h=-4,k = - 7,a = 3,b = 2\). So the equation is \(\frac{(x + 4)^2}{4}+\frac{(y + 7)^2}{9}=1\).

Answer:

\(\frac{(x + 4)^2}{4}+\frac{(y + 7)^2}{9}=1\)