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:
$$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad (a > b)$$

Step2: Find $a$ and $b$ values

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

Step3: Locate vertices & co-vertices

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

Step4: Plot points & draw ellipse

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

Answer:

1. Plot the vertices at $(0, 3)$ and $(0, -3)$, and co-vertices at $(2, 0)$ and $(-2, 0)$ on the provided coordinate grid.
2. Draw a smooth, symmetric ellipse passing through all four plotted points, centered at the origin $(0,0)$ with a vertical major axis.