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 \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\) (since \(a^2 = 9\) and \(b^2=4\), and \(a>b\), it is a vertical ellipse). For a vertical ellipse, the major axis is along the \(y\)-axis, the center is at \((0,0)\) (the origin), the length of the semi - major axis \(a=\sqrt{9} = 3\) and the length of the semi - minor axis \(b=\sqrt{4}=2\).

Step2: Find the vertices and co - vertices

• Vertices: For a vertical ellipse centered at the origin, the vertices are at \((0,\pm a)\). Substituting \(a = 3\), the vertices are \((0,3)\) and \((0, - 3)\).
• Co - vertices: For a vertical ellipse centered at the origin, the co - vertices are at \((\pm b,0)\). Substituting \(b = 2\), the co - vertices are \((2,0)\) and \((- 2,0)\).

Step3: Plot the points and draw the ellipse

• Plot the center \((0,0)\), the vertices \((0,3)\), \((0,-3)\) and the co - vertices \((2,0)\), \((-2,0)\) on the coordinate plane.
• Then, draw a smooth curve connecting these points to form the ellipse. The ellipse will be wider along the \(y\) - axis (since the semi - major axis is along the \(y\) - axis with length 3) and narrower along the \(x\) - axis (with semi - minor axis length 2).

Answer:

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

1. Recognize it as a vertical ellipse centered at \((0,0)\) with \(a = 3\) (semi - major axis, along \(y\) - axis) and \(b=2\) (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. The graph is an ellipse centered at the origin, stretching 3 units up and down from the center (along \(y\) - axis) and 2 units left and right from the center (along \(x\) - axis).