Question

graph each equation.

1. \\(\dfrac{x^2}{4} + \dfrac{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 \(a>b\), 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 \((0,0)\). The vertices (end - points of the major axis) are at \((0,\pm a)\) and the co - vertices (end - points of the minor axis) are at \((\pm b,0)\).
Since \(a=\sqrt{9} = 3\) and \(b=\sqrt{4}=2\), the vertices are \((0, 3)\) and \((0,- 3)\), and the co - vertices are \((2,0)\) and \((-2,0)\).

Step3: Plot the points and draw the ellipse

• Plot the center \((0,0)\).
• Plot the vertices \((0,3)\) and \((0, - 3)\).
• Plot the co - vertices \((2,0)\) and \((-2,0)\).
• Then, draw a smooth curve connecting these points to form the ellipse. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis.

To graph the ellipse:

1. Mark the center at the origin \((0,0)\).
2. Move 2 units to the right and left of the center along the \(x\) - axis to mark the co - vertices \((2,0)\) and \((-2,0)\).
3. Move 3 units up and down from the center along the \(y\) - axis to mark the vertices \((0,3)\) and \((0, - 3)\).
4. Draw a smooth oval - shaped curve passing through these four points, symmetric about both 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, with a major axis along the \(y\) - axis of length \(2a = 6\) (from \(y=-3\) to \(y = 3\)) and a minor axis along the \(x\) - axis of length \(2b=4\) (from \(x=-2\) to \(x = 2\)).)

Answer:

The graph is an ellipse centered at \((0,0)\), with vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\), drawn by plotting the center, vertices, and co - vertices and then drawing a smooth curve through them.