Question

graph each equation.

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

Explanation:

Step1: Identify ellipse standard form

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

Step2: Calculate semi-axes lengths

Solve for $a$ and $b$:
$a = \sqrt{9} = 3$, $b = \sqrt{4} = 2$

Step3: Find intercept points

• x-intercepts: Set $y=0$:

$\frac{x^2}{4} = 1 \implies x^2=4 \implies x=\pm2$
Points: $(-2,0)$ and $(2,0)$

• y-intercepts: Set $x=0$:

$\frac{y^2}{9} = 1 \implies y^2=9 \implies y=\pm3$
Points: $(0,3)$ and $(0,-3)$

Step4: Plot and connect points

Plot the 4 intercepts on the grid, then draw a smooth closed curve through them to form the ellipse.

Answer:

The graph is a vertical ellipse centered at the origin $(0,0)$ with x-intercepts at $(-2,0)$ and $(2,0)$, y-intercepts at $(0,3)$ and $(0,-3)$, forming a smooth closed curve through these points.