Question

graph each equation.

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

coordinate plane with x from -8 to 8, 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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (where \(a>b>0\) and the major axis is along the \(y\)-axis). Here, \(a^{2}=9\) so \(a = 3\), and \(b^{2}=4\) so \(b = 2\).

Step2: Find the vertices and co - vertices

• For the ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), the vertices (endpoints of the major axis) are at \((0,\pm a)=(0,\pm3)\) and the co - vertices (endpoints of the minor axis) are at \((\pm b,0)=(\pm2,0)\).

Step3: Plot the points

• Plot the points \((0, 3)\), \((0,- 3)\), \((2,0)\) and \((-2,0)\) on the coordinate plane.

Step4: Draw the ellipse

Connect the plotted points smoothly to form the ellipse. The ellipse will be centered at the origin \((0,0)\), with the major axis along the \(y\)-axis (vertical major axis) since \(a^{2}\) is under the \(y^{2}\) term. The length of the major axis is \(2a=6\) and the length of the minor axis is \(2b = 4\).

To graph the ellipse:

1. Locate the center at \((0,0)\).
2. Mark the vertices at \((0,3)\) and \((0, - 3)\) (since these are the endpoints of the major axis).
3. Mark the co - vertices at \((2,0)\) and \((-2,0)\) (since these are the endpoints of the minor axis).
4. Draw a smooth curve connecting these points to form the ellipse.

(Note: Since the problem is about graphing, the final answer is the graph of the ellipse with center at \((0,0)\), 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 smoothly.