Question

graph each equation.

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

Explanation:

Step1: Identify the form of the equation

The equation $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is in the standard form of an ellipse $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1$ ($a>b>0$). Here, $a^{2}=9$, so $a = 3$; $b^{2}=4$, so $b=2$.

Step2: Determine the vertices and co - vertices

For an ellipse of the form $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$, the vertices are $(0,\pm a)$ and the co - vertices are $(\pm b,0)$.
The vertices are $(0,3)$ and $(0, - 3)$ (since $a = 3$), and the co - vertices are $(2,0)$ and $(-2,0)$ (since $b = 2$).

Step3: Plot the points and draw the ellipse

Plot the vertices $(0,3)$, $(0, - 3)$, $(2,0)$ and $(-2,0)$ on the coordinate plane. Then, sketch the ellipse passing through these four points.

Answer:

Plot the points $(0,3)$, $(0, - 3)$, $(2,0)$ and $(-2,0)$ and draw an ellipse passing through them.