Question

graph each equation.

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

graph with x-axis from -8 to 8 and y-axis 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 \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\) (where \(a > b\) for a vertical major axis). Here, \(a^2=9\) so \(a = 3\), and \(b^2 = 4\) so \(b=2\). The center of the ellipse is at \((0,0)\) (the origin) since there are no shifts in \(x\) or \(y\) (the numerators are \(x^2\) and \(y^2\) without any linear terms).

Step2: Find the vertices and co - vertices

• For the major axis (along the \(y\) - axis, since \(a\) is under \(y^2\)): The vertices are at \((0,\pm a)=(0,\pm3)\).
• For the minor axis (along the \(x\) - axis): The co - vertices are at \((\pm b,0)=(\pm2,0)\).

Step3: Plot the points

• Plot the center \((0,0)\).
• Plot the vertices \((0,3)\) and \((0, - 3)\).
• Plot the co - vertices \((2,0)\) and \((-2,0)\).

Step4: Draw the ellipse

Connect the plotted points smoothly to form the ellipse. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis, with the major axis along the \(y\) - axis (longer axis) and minor axis along the \(x\) - axis (shorter axis).

Answer:

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

1. Recognize it is an ellipse with center \((0,0)\), \(a = 3\) (semi - major axis along \(y\) - axis), \(b = 2\) (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, symmetric about the \(x\) and \(y\) axes.