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, and a magnifying glass icon near (3,3) or similar

Explanation:

Step1: Identify the ellipse standard form

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 \(a > b\)), centered at the origin \((0,0)\). Here, \(a^2=9\) so \(a = 3\), and \(b^2 = 4\) so \(b=2\).

Step2: Find the vertices and co - vertices

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

Step3: Plot the points

• Plot the vertices \((0, 3)\), \((0,- 3)\) and the co - vertices \((2,0)\), \((-2,0)\).
• Then, sketch the ellipse by connecting these points smoothly, making sure it is 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 \((2,0)\), \((-2,0)\), and it is symmetric about the \(x\) - axis and \(y\) - axis. (The actual graphing involves plotting the points and drawing the ellipse through them as described in the steps above.)