graph each equation.\n9) $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$

Explanation:

Step1: Identify the ellipse form

The equation $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is in the standard - form of an ellipse $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1$ ($a>b>0$), where $a = 3$ and $b = 2$.

Step2: Find the x - intercepts

Set $y = 0$ in the equation $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$. We get $\frac{x^{2}}{4}=1$, then $x^{2}=4$, so $x=\pm2$. The x - intercepts are $(2,0)$ and $(-2,0)$.

Step3: Find the y - intercepts

Set $x = 0$ in the equation $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$. We get $\frac{y^{2}}{9}=1$, then $y^{2}=9$, so $y=\pm3$. The y - intercepts are $(0,3)$ and $(0, - 3)$.

Step4: Sketch the ellipse

Plot the x - intercepts $(2,0),(-2,0)$ and y - intercepts $(0,3),(0, - 3)$ on the coordinate plane and draw a smooth curve to form an ellipse centered at the origin $(0,0)$.

Answer:

Sketch an ellipse centered at the origin with x - intercepts at $x = 2$ and $x=-2$, and y - intercepts at $y = 3$ and $y=-3$.