Question

graph each equation. 9) \\(\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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (where \(a>b>0\)), which is a vertical ellipse centered at the origin \((0,0)\).

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

For the equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\), we have \(b^{2} = 4\) so \(b=\sqrt{4} = 2\) and \(a^{2}=9\) so \(a=\sqrt{9}=3\).

Step3: Determine the vertices and co - vertices

• For a vertical ellipse centered at the origin, the vertices are at \((0,\pm a)=(0,\pm3)\) and the co - vertices are at \((\pm b,0)=(\pm2,0)\).

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:

1. Mark the center at \((0,0)\).
2. Mark the points \((0,3)\), \((0, - 3)\), \((2,0)\) and \((-2,0)\) on the coordinate plane.
3. Draw a smooth elliptical curve passing through these four points. The major axis is along the \(y\) - axis with length \(2a = 6\) and the minor axis is along the \(x\) - axis with length \(2b=4\).

Answer:

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 \(a>b>0\)), which is a vertical ellipse centered at the origin \((0,0)\).

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

For the equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\), we have \(b^{2} = 4\) so \(b=\sqrt{4} = 2\) and \(a^{2}=9\) so \(a=\sqrt{9}=3\).

Step3: Determine the vertices and co - vertices

• For a vertical ellipse centered at the origin, the vertices are at \((0,\pm a)=(0,\pm3)\) and the co - vertices are at \((\pm b,0)=(\pm2,0)\).

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:

1. Mark the center at \((0,0)\).
2. Mark the points \((0,3)\), \((0, - 3)\), \((2,0)\) and \((-2,0)\) on the coordinate plane.
3. Draw a smooth elliptical curve passing through these four points. The major axis is along the \(y\) - axis with length \(2a = 6\) and the minor axis is along the \(x\) - axis with length \(2b=4\).