graph each equation.\n9) \\( \\frac { x ^ { 2 } } { 4 } + \\frac { y ^ { 2 } } { 9 } = 1 \\)

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), so it is a vertical ellipse).

Step2: Find the vertices and co - vertices

For a vertical ellipse (\frac{x^2}{b^2}+\frac{y^2}{a^2}=1), the center is at ((0,0)) (the origin).

• The length of the semi - major axis (a=\sqrt{9} = 3), so the vertices are at ((0,\pm a)=(0, 3)) and ((0,- 3))? Wait, no, (a^2 = 9) so (a = 3), and the vertices are ((0,\pm a)=(0,3)) and ((0, - 3))? Wait, no, wait: the standard form for a vertical ellipse (major axis along y - axis) is (\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1), where ((h,k)) is the center. Here (h = 0,k = 0), (a=\sqrt{9}=3), (b=\sqrt{4} = 2). So the vertices (endpoints of the major axis) are ((0,k\pm a)=(0,0\pm3)=(0,3)) and ((0, - 3))? Wait, no, that's wrong. Wait, if the major axis is along the y - axis, then the vertices are ((0,\pm a)) and the co - vertices are ((\pm b,0)). So (a = 3), so vertices are ((0,3)) and ((0, - 3))? Wait, no, (a^2=9), so (a = 3), so the distance from the center to the vertices along the y - axis is (a = 3), so vertices are ((0,3)) and ((0, - 3))? Wait, no, that can't be. Wait, no, the standard equation of an ellipse with center ((h,k)), major axis length (2a) (along y - axis) and minor axis length (2b) (along x - axis) is (\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1). So when (x = 0), (y=\pm a), and when (y = 0), (x=\pm b). So when (y = 0), (\frac{x^2}{4}=1), so (x^2=4), (x=\pm2). When (x = 0), (\frac{y^2}{9}=1), so (y^2 = 9), (y=\pm3). So the co - vertices (endpoints of the minor axis) are ((\pm2,0)) and the vertices (endpoints of the major axis) are ((0,\pm3)).

Step3: Plot the points

• Plot the center ((0,0)).
• Plot the co - vertices ((2,0)) and ((- 2,0)).
• Plot the vertices ((0,3)) and ((0, - 3)).
• Then draw an ellipse passing through these points. The ellipse will be symmetric about both the x - axis and y - axis.

To graph the ellipse:

1. Mark the center at ((0,0)).
2. Mark the points ((2,0)), ((- 2,0)), ((0,3)) and ((0, - 3)).
3. Sketch a smooth curve connecting these points, making sure the ellipse is wider along the y - axis (since (a = 3) and (b = 2), the major axis is vertical) and symmetric with respect to both the x - axis and y - axis.

(Note: Since the problem asks to graph the equation, the final answer is the graph of the ellipse with center at the origin, vertices at ((0,\pm3)) and co - vertices at ((\pm2,0)) as described above.)

Answer:

The graph is an ellipse centered at the origin ((0,0)) with vertices at ((0, 3)), ((0,-3)) and co - vertices at ((2,0)), ((- 2,0)). The ellipse is drawn by connecting these points with a smooth curve, symmetric about both the (x) - axis and (y) - axis.