Question

graph each equation.

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

Explanation:

Step1: Identify ellipse standard form

The equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ ($a>b$) is a vertical ellipse centered at $(0,0)$.
Here, $a^2=9 \implies a=3$, $b^2=4 \implies b=2$.

Step2: Find vertices and co-vertices

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

Step3: Plot key points and draw ellipse

Mark the 4 key points on the grid, then sketch a smooth closed curve through them.

Answer:

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