Question

graph each equation.

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

Explanation:

Step1: Identify ellipse parameters

The standard form of a vertical ellipse is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $a > b$. For $\frac{x^2}{4} + \frac{y^2}{9} = 1$, we have $a^2=9 \implies a=3$, $b^2=4 \implies b=2$.

Step2: Find vertices and co-vertices

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

Step3: Plot key points and draw ellipse

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

Answer:

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