graph each equation.\n9) \\(\frac{x^2}{4} + \frac{y^2}{9} = 1\\)\ngraph with x-axis from -8 to 8 and y-axis…

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^2 = 9) and (b^2=4), and (a>b), it is a vertical ellipse). For a vertical ellipse, the major axis is along the (y)-axis, the center is at ((0,0)) (the origin), the length of the semi - major axis (a=\sqrt{9} = 3) and the length of the semi - minor axis (b=\sqrt{4}=2).

Step2: Find the vertices and co - vertices

• Vertices: For a vertical ellipse centered at the origin, the vertices are at ((0,\pm a)). Substituting (a = 3), the vertices are ((0,3)) and ((0, - 3)).
• Co - vertices: For a vertical ellipse centered at the origin, the co - vertices are at ((\pm b,0)). Substituting (b = 2), the co - vertices are ((2,0)) and ((- 2,0)).

Step3: Plot the points and draw the ellipse

• Plot the center ((0,0)), the vertices ((0,3)), ((0,-3)) and the co - vertices ((2,0)), ((-2,0)) on the coordinate plane.
• Then, draw a smooth curve connecting these points to form the ellipse. The ellipse will be wider along the (y) - axis (since the semi - major axis is along the (y) - axis with length 3) and narrower along the (x) - axis (with semi - minor axis length 2).

Answer:

To graph (\boldsymbol{\frac{x^2}{4}+\frac{y^2}{9}=1}):

1. Recognize it as a vertical ellipse centered at ((0,0)) with (a = 3) (semi - major axis, along (y) - axis) and (b=2) (semi - minor axis, along (x) - axis).
2. Plot vertices ((0,3)), ((0, - 3)) and co - vertices ((2,0)), ((-2,0)).
3. Draw a smooth ellipse through these points. The graph is an ellipse centered at the origin, stretching 3 units up and down from the center (along (y) - axis) and 2 units left and right from the center (along (x) - axis).