Graphing a Half-Ellipse

Example

Graph  $\frac{y}{4}=\sqrt[]{1-\frac{x^2}{25}}$. Give the domain and range.

Solution

Square each side to get $\frac{y^2}{16}=1-\frac{x^2}{25}$, or $\frac{x^2}{25}+\frac{y^2}{16}=1.$

This is the equation of an ellipse with x-intercepts $\pm5$ and y-intercepts $\left(0,\pm4\right)$.

In the original equation, the radical expression $\sqrt[]{1-\frac{x^2}{25}}$ represents a nonnegative number, so the only possible values of y are those that give the half-ellipse shown below.

This is the graph of a function with domain $\left[-5,5\right]$ and range $\left[0,4\right]$.