Question

graph each equation.

1. \\(\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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (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 (vertical, along the \(y\) - axis), the vertices are at \((0,\pm a)=(0,\pm3)\).
• For the minor axis (horizontal, along the \(x\) - axis), the co - vertices are at \((\pm b,0)=(\pm2,0)\).

Step3: Plot the points and draw the ellipse

Plot the points \((0,3)\), \((0, - 3)\), \((2,0)\), and \((- 2,0)\) on the coordinate plane. Then, draw a smooth ellipse passing through these points. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis.

Answer:

The graph is an ellipse centered at the origin with vertices at \((0,3)\), \((0, - 3)\) and co - vertices at \((2,0)\), \((- 2,0)\), drawn by plotting these points and sketching a smooth curve through them. (Note: Since the question asks to graph, the final answer is the description of the graph construction as above. If a visual representation was to be provided, it would be an ellipse with the mentioned key points.)