Question

graph each equation.

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

Explanation:

Step1: Identify ellipse standard form

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

Step2: Find $a$ and $b$ values

$a^2=9 \implies a=3$, $b^2=4 \implies b=2$

Step3: Locate key vertices

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

Step4: Plot and connect points

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

Answer:

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