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 type

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 \(a>b>0\)), which is a vertical ellipse centered at the origin \((0,0)\).

Step2: Find the values of \(a\) and \(b\)

For the given equation, compare with \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\). We have \(a^{2}=9\) so \(a = 3\) and \(b^{2}=4\) so \(b = 2\).

Step3: Determine the vertices and co - vertices

• The vertices of a vertical ellipse centered at the origin are \((0,\pm a)=(0,\pm3)\).
• The co - vertices are \((\pm b,0)=(\pm2,0)\).

Step4: Plot the points and draw the ellipse

Plot the points \((0, 3)\), \((0,- 3)\), \((2,0)\) and \((-2,0)\). Then draw a smooth curve connecting these points to form the ellipse. The major axis is along the \(y\) - axis with length \(2a = 6\) and the minor axis is along the \(x\) - axis with length \(2b=4\).

Answer:

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

1. Recognize it as a vertical ellipse centered at \((0,0)\) with \(a = 3\) (semi - major axis) and \(b=2\) (semi - minor axis).
2. Plot the vertices \((0,3)\), \((0, - 3)\) and co - vertices \((2,0)\), \((-2,0)\).
3. Draw a smooth elliptical curve passing through these points. The ellipse will be taller along the \(y\) - axis (since \(a>b\)) and symmetric about both the \(x\) - axis and \(y\) - axis.