Question

graph each equation.

1. $\frac{x^2}{4} + \frac{y^2}{9} = 1$

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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (since \(a^{2}=9\) and \(b^{2} = 4\), and \(a>b\), so it is a vertical ellipse).

Step2: Find the vertices and co - vertices

For a vertical ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), the center is at \((0,0)\) (since there are no \(h\) or \(k\) shifts in the equation).

• The length of the semi - major axis \(a=\sqrt{9}=3\), so the vertices are at \((0,\pm a)=(0, 3)\) and \((0,- 3)\).
• The length of the semi - minor axis \(b = \sqrt{4}=2\), so the co - vertices are at \((\pm b,0)=(\pm2,0)\).

Step3: Plot the points

Plot the center \((0,0)\), the vertices \((0,3)\), \((0, - 3)\) and the co - vertices \((2,0)\), \((- 2,0)\). Then, draw a smooth curve connecting these points to form the ellipse.

(Note: Since the problem asks to graph the equation, the key steps are identifying the type of conic, finding the important points (center, vertices, co - vertices) and then plotting them to form the graph. Here we have described the steps to graph the ellipse represented by the equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\))

Answer:

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

1. Recognize it is a vertical ellipse (center at \((0,0)\)) with \(a = 3\) (semi - major axis) and \(b=2\) (semi - minor 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 along the \(y\) - axis and 2 units left and right along the \(x\) - axis.