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

Step2: Find $a$ and $b$ values

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

Step3: Locate 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

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

Answer:

The graph is an ellipse with vertices at $(2,0)$, $(-2,0)$, $(0,3)$, and $(0,-3)$, forming a vertical elongated oval centered at the origin.