Question

graph each equation.

1. $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$

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: Calculate axis lengths

For $a^2=9$, $a=3$; for $b^2=4$, $b=2$.

Step3: Find key points

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

Step4: Plot and connect points

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

Answer:

The graph is an ellipse centered at the origin $(0,0)$ with vertices at $(0, 3)$, $(0, -3)$ and co-vertices at $(2, 0)$, $(-2, 0)$, forming a vertically elongated oval shape passing through these four points.