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 form $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $a > b$.

Step2: Calculate $a$ and $b$

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

Step3: Find key points

Vertices (on y-axis): $(0, \pm a) = (0, \pm 3)$
Co-vertices (on x-axis): $(\pm b, 0) = (\pm 2, 0)$

Step4: Plot and draw ellipse

Plot the 4 key points, then sketch a smooth closed curve connecting them to form the ellipse.

Answer:

An ellipse with vertices at $(0, 3)$, $(0, -3)$ and co-vertices at $(2, 0)$, $(-2, 0)$