Question

graph each equation.\

1. \\(\dfrac{x^2}{4} + \dfrac{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 $(0,0)$.
Given $\frac{x^2}{4} + \frac{y^2}{9} = 1$, so $a^2=9$, $b^2=4$.

Step2: Calculate semi-axes lengths

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

Step3: Find key points

• Vertices (vertical): $(0, \pm a) = (0, 3), (0, -3)$
• Co-vertices (horizontal): $(\pm b, 0) = (2, 0), (-2, 0)$

Step4: Plot and draw ellipse

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

Answer:

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