Question

graph each equation.

1. \\(\dfrac{x^2}{4} + \dfrac{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

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 \(a>b>0\)), so it is an ellipse centered at the origin \((0,0)\).

Step2: Find the semi - axes lengths

For the ellipse equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\), we compare with \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\). We have \(a^{2}=9\) and \(b^{2}=4\), so \(a = 3\) and \(b = 2\). The major axis is along the \(y\) - axis (since \(a\) is associated with the \(y\) - term) and the minor axis is along the \(x\) - axis.

Step3: Find the vertices and co - vertices

• Vertices: The vertices of the ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) are \((0,\pm a)\). Substituting \(a = 3\), the vertices are \((0,3)\) and \((0, - 3)\).
• Co - vertices: The co - vertices of the ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) are \((\pm b,0)\). Substituting \(b = 2\), the co - vertices are \((2,0)\) and \((- 2,0)\).

Step4: Plot the points and draw the ellipse

• Plot the points \((0,3)\), \((0,-3)\), \((2,0)\) and \((- 2,0)\) on the coordinate plane.
• Then, sketch an ellipse passing through these points. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis.

To graph the ellipse:

1. Mark the center at \((0,0)\).
2. Move 3 units up and down from the center along the \(y\) - axis to get the vertices \((0,3)\) and \((0, - 3)\).
3. Move 2 units left and right from the center along the \(x\) - axis to get the co - vertices \((-2,0)\) and \((2,0)\).
4. Draw a smooth curve connecting these points to form the ellipse.

(Note: Since the problem is about graphing, the final answer is the graph of the ellipse with center at the origin, vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\) as described above.)

Answer:

The graph is an ellipse centered at \((0,0)\) with vertices \((0, 3)\), \((0, - 3)\) and co - vertices \((2, 0)\), \((-2, 0)\) (the ellipse is drawn through these points on the given coordinate grid).