Question

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

Explanation:

Step1: Identify the center

The center of the ellipse is at the point $(3,4)$ as seen from the graph.

Step2: Determine the semi - major and semi - minor axes

The ellipse extends 3 units horizontally from the center (from $x = 0$ to $x=6$ relative to the center at $x = 3$), so $a = 3$. It extends 2 units vertically from the center (from $y = 2$ to $y = 6$ relative to the center at $y=4$), so $b = 2$.

Step3: Write the standard form of the ellipse equation

The standard form of an ellipse with center $(h,k)$ is $\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1$. Substituting $h = 3,k = 4,a = 3,b = 2$ we get $\frac{(x - 3)^2}{9}+\frac{(y - 4)^2}{4}=1$.

Answer:

$\frac{(x - 3)^2}{9}+\frac{(y - 4)^2}{4}=1$