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, origin at (0,0)

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 \(a > b>0\)) for an ellipse centered at the origin with a vertical major axis. Here, \(a^{2}=9\) so \(a = 3\) and \(b^{2}=4\) so \(b = 2\).

Step2: Find the vertices and co - vertices

• For the major axis (along the \(y\) - axis, since \(a\) is associated with \(y^{2}\)): 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 vertices \((0, 3)\) and \((0,- 3)\) on the \(y\) - axis.
• Plot the co - vertices \((2,0)\) and \((- 2,0)\) on the \(x\) - axis.

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.

Answer:

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

1. Recognize it as an ellipse centered at the origin with a vertical major axis.
2. Identify vertices \((0,\pm3)\) and co - vertices \((\pm2,0)\).
3. Plot the points \((0,3)\), \((0, - 3)\), \((2,0)\), \((-2,0)\).
4. Draw a smooth ellipse passing through these points, symmetric about the \(x\) - axis and \(y\) - axis.