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

Calculate semi-axes:
$a = \sqrt{9} = 3$, $b = \sqrt{4} = 2$

Step3: Locate key vertices

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

Step4: Plot and connect points

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

Answer:

The graph is a vertical ellipse with vertices at $(0, 3)$, $(0, -3)$, $(2, 0)$, $(-2, 0)$, forming a smooth closed curve passing through these points on the provided coordinate grid.