Question

graph each equation. 9) \\( \frac{x^2}{4} + \frac{y^2}{9} = 1 \\) graph with x from -8 to 8, y from -8 to 8, grid lines, axes

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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (where \(a > b>0\)) for an ellipse centered at the origin with a vertical major axis.

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

For the given equation, we have \(a^{2}=9\) and \(b^{2} = 4\). So, \(a=\sqrt{9}=3\) and \(b=\sqrt{4} = 2\).

Step3: Find the vertices and co - vertices

• The vertices of the ellipse (endpoints of the major axis, along the \(y\) - axis) are \((0,\pm a)=(0,\pm3)\).
• The co - vertices of the ellipse (endpoints of the minor axis, along the \(x\) - axis) are \((\pm b,0)=(\pm2,0)\).

Step4: Plot the points

• Plot 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.

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

1. Recognize it is an ellipse centered at the origin \((0,0)\) with a vertical major axis (since the denominator under \(y^{2}\) is larger).
2. The length of the semi - major axis \(a = 3\) (distance from center to vertices on \(y\) - axis), so vertices are \((0,3)\) and \((0, - 3)\).
3. The length of the semi - minor axis \(b=2\) (distance from center to co - vertices on \(x\) - axis), so co - vertices are \((2,0)\) and \((-2,0)\).
4. Plot the points \((0,3)\), \((0, - 3)\), \((2,0)\), \((-2,0)\) and sketch the ellipse passing through these points.

(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, with vertices at \((0,3)\) and \((0, - 3)\) and co - vertices at \((2,0)\) and \((-2,0)\))

Answer:

The graph is an ellipse centered at the origin with vertices \((0,\pm3)\) and co - vertices \((\pm2,0)\) (and the ellipse is drawn through these points as described in the steps above).