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, 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). Here, the center of the ellipse is at the origin \((0,0)\) because there are no shifts in \(x\) or \(y\) (the numerators are \(x^{2}\) and \(y^{2}\) without any linear terms).

Step2: Find the vertices and co - vertices

For a vertical ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), the vertices are at \((0,\pm a)\) and the co - vertices are at \((\pm b,0)\).

• Calculate \(a\) and \(b\):
• Given \(a^{2}=9\), then \(a=\sqrt{9} = 3\). So the vertices are \((0, 3)\) and \((0,- 3)\).
• Given \(b^{2}=4\), then \(b=\sqrt{4}=2\). So the co - vertices are \((2,0)\) and \((- 2,0)\).

Step3: Plot the center, vertices, and co - vertices

• The center is \((0,0)\).
• Plot the vertices \((0,3)\) (on the positive \(y\) - axis, 3 units up from the origin) and \((0, - 3)\) (on the negative \(y\) - axis, 3 units down from the origin).
• Plot the co - vertices \((2,0)\) (on the positive \(x\) - axis, 2 units to the right of the origin) and \((-2,0)\) (on the negative \(x\) - axis, 2 units to the left of the origin).

Step4: Sketch the ellipse

Draw a smooth curve that passes through the vertices and co - vertices. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis (since the center is at the origin and the equation is symmetric in \(x\) and \(y\) (replacing \(x\) with \(-x\) or \(y\) with \(-y\) does not change the equation)).

To graph the ellipse:

1. Mark the center \((0,0)\).
2. Mark the vertices \((0,3)\) and \((0, - 3)\).
3. Mark the co - vertices \((2,0)\) and \((-2,0)\).
4. Draw a smooth oval - shaped curve (ellipse) connecting these points, making sure the curve is symmetric about both the \(x\) - axis and \(y\) - axis.

(Note: Since the problem asks to graph the equation, the above steps describe how to plot the key points and draw the ellipse. If we were to represent the graph in a textual description of the plotting process, the steps above outline the construction. If we were to use a graphing tool, we would plot the points and draw the ellipse as described.)

Answer:

The graph is an ellipse centered at the origin \((0,0)\) with vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\), and a smooth curve connecting these points symmetric about the \(x\) - axis and \(y\) - axis. (To actually draw it, plot the center, vertices, co - vertices and draw the ellipse as per the steps above.)