Question

graph each equation.

1. \\(dfrac{x^2}{4} + dfrac{y^2}{9} = 1\\)

Explanation:

Step1: Identify ellipse standard form

The equation $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (where $a > b$) is a vertical ellipse centered at $(0,0)$.
Compare to $\frac{x^2}{4} + \frac{y^2}{9} = 1$, so $a^2=9$, $b^2=4$.

Step2: Calculate semi-axes lengths

Solve for $a$ and $b$:
$a = \sqrt{9} = 3$, $b = \sqrt{4} = 2$

Step3: Find intercept points

• x-intercepts: Set $y=0$: $\frac{x^2}{4}=1 \implies x=\pm2$, so points $(-2,0)$ and $(2,0)$.
• y-intercepts: Set $x=0$: $\frac{y^2}{9}=1 \implies y=\pm3$, so points $(0,3)$ and $(0,-3)$.

Step4: Plot and draw the ellipse

Plot the 4 intercept points, then draw a smooth closed curve connecting them, centered at the origin, stretched taller along the y-axis.

Answer:

The graph is a vertical ellipse centered at $(0,0)$ with x-intercepts at $(-2,0)$ and $(2,0)$, y-intercepts at $(0,3)$ and $(0,-3)$, forming a smooth closed curve through these points.