graph each equation.\n\n9) $\\frac{x^{2}}{4} + \\frac{y^{2}}{9} = 1$

Answer:

The graph is a vertical ellipse centered at $(0, 0)$ with vertices at $(0, 3)$ and $(0, -3)$, and co-vertices at $(2, 0)$ and $(-2, 0)$.

Explanation:

Step1: Identify the conic section type

The equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ represents an ellipse centered at the origin $(0, 0)$.

Step2: Determine the semi-axes lengths

Compare to $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$: $$a^2 = 9 \Rightarrow a = 3$$ $$b^2 = 4 \Rightarrow b = 2$$

Step3: Locate the vertices and co-vertices

Since $a > b$, the major axis is vertical. Vertices: $(0, \pm 3)$ Co-vertices: $(\pm 2, 0)$

Step4: Sketch the graph

Plot the four points $(0, 3)$, $(0, -3)$, $(2, 0)$, and $(-2, 0)$ and connect them with a smooth curve.