Question

graph each equation.

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

Explanation:

Step1: Identify ellipse standard form

The given equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ matches the vertical major axis ellipse standard form $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $a > b$.

Step2: Calculate $a$ and $b$

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

Step3: Find intercept points

• x-intercepts: Set $y=0$: $\frac{x^2}{4}=1 \implies x=\pm2$, so points $(-2,0)$ and $(2,0)$.
• y-intercepts: Set $x=0$: $\frac{y^2}{9}=1 \implies y=\pm3$, so points $(0,3)$ and $(0,-3)$.

Step4: Plot and connect points

Plot the 4 intercepts, then draw a smooth closed curve through them.

Answer:

The graph is an ellipse with x-intercepts at $(-2, 0)$ and $(2, 0)$, y-intercepts at $(0, 3)$ and $(0, -3)$, and a vertical major axis. When plotted on the grid, it is a smooth oval curve passing through these four points.