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 \(a>b\) and major axis is along y - axis), where \(a^2 = 9\) and \(b^2=4\). So, \(a = 3\) and \(b = 2\).

Step2: Find the vertices and co - vertices

• For the ellipse \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\), the vertices (end - points of the major axis) are at \((0,\pm a)=(0,\pm3)\) and the co - vertices (end - points 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.
• Then, draw a smooth curve connecting these points to form the ellipse. The ellipse will be centered at the origin \((0,0)\) with the major axis along the y - axis (since the denominator of \(y^2\) is larger) and minor axis along the x - axis.

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

1. Recognize it is an ellipse with center \((0,0)\), \(a = 3\) (semi - major axis along y - axis), \(b = 2\) (semi - minor axis along x - axis).
2. Plot the vertices \((0,3)\), \((0, - 3)\) and co - vertices \((2,0)\), \((- 2,0)\).
3. Draw a smooth ellipse passing through these points.

(Note: Since the question is about graphing, the final answer is the graph of the ellipse with the above - mentioned properties. If we were to describe the key points for graphing: vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\) and the ellipse centered at the origin with major axis vertical.)

Answer:

The graph is an ellipse centered at the origin \((0,0)\) with vertices at \((0, 3)\), \((0,-3)\) and co - vertices at \((2,0)\), \((-2,0)\). To draw it, plot these four points and sketch a smooth curve connecting them, with the major axis along the \(y\) - axis and minor axis along the \(x\) - axis.