Question

graph each equation.

1. \\(\frac{x^2}{4} + \frac{y^2}{9} = 1\\)

Explanation:

Step1: Identify ellipse standard form

The equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ (since $a>b$) is a vertical ellipse centered at $(0,0)$, where $a^2=9$, $b^2=4$.

Step2: Calculate $a$ and $b$

$a=\sqrt{9}=3$, $b=\sqrt{4}=2$

Step3: Find key vertices

• Vertical vertices: $(0, \pm a)=(0,3), (0,-3)$
• Horizontal vertices: $(\pm b, 0)=(2,0), (-2,0)$

Step4: Plot and draw ellipse

Mark the 4 vertices on the grid, then draw a smooth closed curve through them.

Answer:

The graph is a vertical ellipse centered at the origin $(0,0)$ with vertices at $(2,0)$, $(-2,0)$, $(0,3)$, and $(0,-3)$, forming a smooth oval shape passing through these points.