Question

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

Explanation:

Step1: Find the center of the ellipse

The center of the ellipse \((h,k)\) is \((-3,-6)\) (by observing the mid - point of the ellipse on the graph).

Step2: Determine \(a\) and \(b\)

The distance from the center to the right (or left) vertex is \(a\). Counting the units, \(a = 8\). The distance from the center to the top (or bottom) co - vertex is \(b\). Counting the units, \(b = 2\).

Step3: Write the standard form of the ellipse equation

The standard form of an ellipse with a horizontal major axis is \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}}=1\).
Substitute \(h=-3,k = - 6,a = 8,b = 2\) into the formula:
\(\frac{(x+3)^{2}}{64}+\frac{(y + 6)^{2}}{4}=1\)

Answer:

\(\frac{(x + 3)^{2}}{64}+\frac{(y+6)^{2}}{4}=1\)