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 \(a>b\), so 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 \((0,0)\) (since there are no shifts in \(x\) or \(y\) in the equation).

• The value of \(a=\sqrt{9} = 3\), so the vertices are at \((0,\pm a)=(0, 3)\) and \((0,- 3)\).
• The value of \(b=\sqrt{4}=2\), so the co - vertices are at \((\pm b,0)=(2,0)\) 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 an ellipse passing through these four points, centered at the origin, with the major axis along the \(y\) - axis (since \(a\) is associated with the \(y\) - term) and the minor axis along the \(x\) - axis.

To graph the ellipse:

1. Mark the center at \((0,0)\).
2. Mark the vertices \((0,3)\) (3 units up from the center on the \(y\) - axis) and \((0, - 3)\) (3 units down from the center on the \(y\) - axis).
3. Mark the co - vertices \((2,0)\) (2 units to the right of the center on the \(x\) - axis) and \((-2,0)\) (2 units to the left of the center on the \(x\) - axis).
4. Sketch a smooth curve connecting these points, forming an ellipse. The ellipse will be taller than it is wide, with the longest diameter (major axis) along the \(y\) - axis and the shortest diameter (minor axis) along the \(x\) - axis.

(Note: Since the problem asks to graph the equation, the final answer is the graphical representation as described above. If we were to describe the key points for plotting: center \((0,0)\), vertices \((0,\pm3)\), co - vertices \((\pm2,0)\) and the ellipse passing through them.)

Answer:

The graph is an ellipse centered at the origin \((0,0)\) with vertices at \((0, 3)\), \((0,-3)\) and co - vertices at \((2,0)\), \((-2,0)\). The ellipse is drawn by connecting these points smoothly, with the major axis along the \(y\) - axis and the minor axis along the \(x\) - axis.