Question

graph each equation.

1. \\(\dfrac{x^2}{4} + \dfrac{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 key vertices

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

Step4: Plot and connect points

Mark the 4 vertices on the grid, 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.