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

Step2: Calculate $a$ and $b$

$a = \sqrt{9} = 3$, $b = \sqrt{4} = 2$

Step3: Find vertices (y-axis)

Vertices are at $(0, \pm a) = (0, 3)$ and $(0, -3)$

Step4: Find co-vertices (x-axis)

Co-vertices are at $(\pm b, 0) = (2, 0)$ and $(-2, 0)$

Step5: Plot points and draw ellipse

Plot the 4 points, then sketch a smooth ellipse through them.

Answer:

This is a vertical ellipse with vertices at $(0, 3)$, $(0, -3)$ and co-vertices at $(2, 0)$, $(-2, 0)$. The graph is a smooth oval curve passing through these four points on the provided coordinate grid.