Question

An equation and its graph are given. Find the x- and y-intercepts.

$\frac{x^2}{9}+\frac{y^2}{4}=1$

Short answer

The x-intercepts are 3 and y-intercepts are and 2.

Step-by-step solution

Step 1. Apply the concept of intercepts.

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on the y-axis where the graph of the equation intersects the y-axis.

Step 2. Compute the x-intercept.

Consider the provided equation$\frac{x^2}{9}+\frac{y^2}{4}=1$and the graph of the equation is sketched below.

Observe from the graph that the curve intersects thex-axis at two points and 3.

Recall that the x-intercepts are the points on thex-axis where the graph of the equation intersects the x-axis.

Substitute $y=0$in the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$.

localid="1643703045737" $\begin{array}{c}\frac{x^2}{9}+\frac{0^2}{4}=1\\ \frac{x^2}{9}=1\\ x^2=9\end{array}$

Either $x=-3$or $x=3$.

Therefore, x-intercepts are $-3$and 3.

Step 3. Compute the y-intercept.

Consider the provided equation$\frac{x^2}{9}+\frac{y^2}{4}=1$ and the graph of the equation is sketched below.

Observe from the graph that the curve intersects they-axis at two points and 2.

Recall that the y-intercepts are the points on they-axis where the graph of the equation intersects the y-axis.

Substitute $x=0$in the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$.

$$\begin{gathered} \frac{0^2}{9}+\frac{y^2}{4}=1 \\ \frac{y^2}{4}=1 \\ {y}^2=4 \end{gathered}$$

Either $y=-2$or $y=2$.

Therefore, x-intercepts are $-2$and 2.

Thus, the x-intercepts are $-3$ and 3 and y-intercepts are $-2$ and 2.