Question

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

Explanation:

Step1: Identify the center of the ellipse

By observing the graph, the center of the ellipse $(h,k)$ is at $(-7,7)$.

Step2: Determine the semi - major axis $a$ and semi - minor axis $b$

Counting the grid squares, the distance from the center to the farthest point horizontally (semi - major axis if the ellipse is horizontally oriented or semi - minor axis if vertically oriented) gives $a = 2$. The distance from the center to the farthest point vertically gives $b= 3$. Since the ellipse is vertically oriented, the standard form of the equation of an ellipse is $\frac{(y - k)^2}{a^{2}}+\frac{(x - h)^2}{b^{2}}=1$.

Step3: Substitute the values of $h$, $k$, $a$ and $b$ into the standard form

Substitute $h=-7$, $k = 7$, $a = 3$ and $b = 2$ into the equation $\frac{(y - k)^2}{a^{2}}+\frac{(x - h)^2}{b^{2}}=1$. We get $\frac{(y - 7)^2}{9}+\frac{(x + 7)^2}{4}=1$.

Answer:

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