Question

graph each equation.

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

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 \(a>b\), 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 shifts in \(x\) and \(y\) from the origin).

• The value of \(a=\sqrt{9} = 3\) and \(b=\sqrt{4}=2\).
• The vertices (end - points of the major axis) are at \((0,\pm a)=(0,\pm3)\).
• The co - vertices (end - points of the minor axis) are at \((\pm b,0)=(\pm2,0)\).

Step3: Plot the points

• Plot the center \((0,0)\).
• Plot the vertices \((0,3)\) and \((0, - 3)\).
• Plot the co - vertices \((2,0)\) and \((-2,0)\).

Step4: Draw the ellipse

Connect the plotted points smoothly to form the ellipse. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis.

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

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

(Note: Since the question asks to graph the equation, the above steps describe how to plot the ellipse. If we were to describe the graph: It 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.)

Answer:

The graph is an ellipse centered at \((0,0)\) with vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\), drawn by connecting these points symmetrically about the \(x\) and \(y\) axes. (The actual graph is a smooth curve passing through \((0,3)\), \((0, - 3)\), \((2,0)\), \((-2,0)\) and symmetric about the coordinate axes.)