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 $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $a > b$.

Step2: Calculate axis lengths

Here, $a^2=9 \Rightarrow a=3$, $b^2=4 \Rightarrow b=2$.

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

Mark the 4 key points on the grid, then draw a smooth closed curve through them.

Answer:

该方程的图像是一个以原点为中心，长轴在y轴上，顶点为$(0, 3)$、$(0, -3)$，短轴顶点为$(2, 0)$、$(-2, 0)$的椭圆，按照上述关键点在给定网格中绘制光滑闭合曲线即可得到图像。