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, origin at (0,0)

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 centered at the origin \((0,0)\)).

Step2: Find the vertices and co - vertices

For a vertical ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (\(a>b>0\)):

• The vertices are at \((0,\pm a)\). Given \(a^{2}=9\), then \(a = 3\), so the vertices are \((0,3)\) and \((0, - 3)\).
• The co - vertices are at \((\pm b,0)\). Given \(b^{2}=4\), then \(b = 2\), so the co - vertices are \((2,0)\) and \((-2,0)\).

Step3: Plot the points and draw the ellipse

1. Plot the center \((0,0)\).
2. Plot the vertices \((0,3)\) and \((0,-3)\) (these are the top - most and bottom - most points of the ellipse).
3. Plot the co - vertices \((2,0)\) and \((-2,0)\) (these are the right - most and left - most points of the ellipse).
4. Then, draw a smooth curve connecting these points to form the ellipse. The major axis (the longer axis) is along the \(y\) - axis with length \(2a=6\) and the minor axis (the shorter axis) is along the \(x\) - axis with length \(2b = 4\).

To graph the ellipse:

• Mark the center at \((0,0)\).
• Mark \((0,3)\), \((0, - 3)\), \((2,0)\), and \((-2,0)\).
• Sketch the ellipse passing through these four points, symmetric about both the \(x\) - axis and \(y\) - axis.

(Note: Since the question asks to graph the equation, the above steps describe how to plot the key points and draw the ellipse. If we were to describe the graph, it is an ellipse centered at the origin, with vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\), symmetric about the \(x\) - axis and \(y\) - axis.)

Answer:

The graph is an ellipse centered at the origin \((0,0)\) with vertices \((0,3)\), \((0, - 3)\) and co - vertices \((2,0)\), \((-2,0)\), symmetric about the \(x\) - axis and \(y\) - axis. (The actual drawing involves plotting the center, vertices, co - vertices and drawing a smooth elliptical curve through them.)