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 $a = \sqrt{9} = 3$, $b = \sqrt{4} = 2$

Step3: Locate vertices and co-vertices

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

Step4: Plot points and draw ellipse

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

Answer:

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