Question

graph each equation.

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

Explanation:

Step1: Identify ellipse parameters

The equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ matches the vertical ellipse standard form $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ ($a>b$). Here, $a^2=9 \Rightarrow a=3$, $b^2=4 \Rightarrow b=2$.

Step2: Find vertices & co-vertices

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

Step3: Plot points & draw ellipse

Mark the 4 points on the grid, then draw a smooth closed curve through them.

Answer:

1. Plot the vertical vertices at $(0, 3)$ and $(0, -3)$, and horizontal co-vertices at $(2, 0)$ and $(-2, 0)$ on the provided coordinate grid.
2. Draw a smooth, symmetrical oval (ellipse) that passes through all four plotted points, centered at the origin $(0,0)$.