graph each equation.\n\n9) $\\frac{x^2}{4} + \\frac{y^2}{9} = 1$
graph each equation.\n\n9) $\\frac{x^2}{4} + \\frac{y^2}{9} = 1$
Answer
Explanation:
Step1: Identify the conic section type
The equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ represents an ellipse centered at $(0,0)$.
Step2: Determine the semi-axes lengths
The denominators are $a^2$ and $b^2$. Since $9 > 4$, the major axis is vertical. $$a^2 = 9 \Rightarrow a = 3, \quad b^2 = 4 \Rightarrow b = 2$$
Step3: Identify the vertices and co-vertices
The vertices are at $(0, \pm a)$ and the co-vertices are at $(\pm b, 0)$. $$\text{Vertices: } (0, 3), (0, -3); \quad \text{Co-vertices: } (2, 0), (-2, 0)$$
Step4: Sketch the graph
Plot the four points and draw a smooth curve connecting them.
Answer:
The graph is an ellipse centered at $(0,0)$ with vertices at $(0, 3)$ and $(0, -3)$, and co-vertices at $(2, 0)$ and $(-2, 0)$.