Question

graph each equation.

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

coordinate plane with x from -8 to 8 and y from -8 to 8, grid lines

Explanation:

Step1: Identify the conic section

The equation $\frac{x^2}{4}+\frac{y^2}{9} = 1$ is in the standard form of an ellipse, $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ (since $a > b$ for vertical major axis), where $a^2 = 9$ and $b^2=4$. So $a = 3$ and $b = 2$.

Step2: Find the vertices and co - vertices

• For the ellipse with vertical major axis, the vertices are at $(0,\pm a)=(0,\pm3)$ and the co - vertices are at $(\pm b,0)=(\pm2,0)$.

Step3: Plot the points

• Plot the vertices: $(0, 3)$ and $(0, - 3)$.
• Plot the co - vertices: $(2,0)$ and $(-2,0)$.

Step4: Sketch the ellipse

Connect the plotted points smoothly to form the ellipse. The ellipse will be centered at the origin $(0,0)$ (since there are no shifts in the $x$ or $y$ terms in the equation), with a vertical major axis (because the denominator under $y^2$ is larger) of length $2a = 6$ and a horizontal minor axis of length $2b=4$.

Answer:

The graph is an ellipse centered at the origin with vertices at $(0, 3)$, $(0, - 3)$ and co - vertices at $(2, 0)$, $(-2, 0)$, and it is sketched by connecting these points smoothly. (To actually draw it on the given grid, plot the four points and draw the ellipse passing through them.)