Question

graph each equation.

1. $\frac{x^2}{4} + \frac{y^2}{9} = 1$

coordinate grid with x from -8 to 8 and y from -8 to 8

Explanation:

Step1: Identify the conic section type

The given equation is \(\frac{x^{2}}{4}+\frac{y^{2}}{9} = 1\), which is in the standard form of an ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (where \(a>b>0\) and the major axis is along the \(y\)-axis).

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

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

Step3: Find the vertices and co - vertices

• The vertices of the ellipse (along the major axis, \(y\)-axis) are \((0,\pm a)=(0,\pm3)\).
• The co - vertices of the ellipse (along the minor axis, \(x\)-axis) are \((\pm b,0)=(\pm2,0)\).

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, draw a smooth curve connecting these points to form the ellipse. The ellipse will be centered at the origin \((0,0)\) since there are no shifts in the \(x\) or \(y\) terms in the equation.

Graph Description:

The graph is an ellipse centered at the origin. It has vertices at \((0, 3)\) and \((0, - 3)\), and co - vertices at \((2,0)\) and \((-2,0)\). To sketch it, plot these four points and draw a smooth, closed curve passing through them, with the major axis along the \(y\) - axis and the minor axis along the \(x\) - axis.

(Note: Since the question asks to graph the equation, the final answer is the graph of the ellipse as described above. If we were to represent the key points: vertices \((0,3)\), \((0, - 3)\) and co - vertices \((2,0)\), \((-2,0)\) are the main points to plot for the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\))

Answer:

Step1: Identify the conic section type

The given equation is \(\frac{x^{2}}{4}+\frac{y^{2}}{9} = 1\), which is in the standard form of an ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (where \(a>b>0\) and the major axis is along the \(y\)-axis).

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

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

Step3: Find the vertices and co - vertices

• The vertices of the ellipse (along the major axis, \(y\)-axis) are \((0,\pm a)=(0,\pm3)\).
• The co - vertices of the ellipse (along the minor axis, \(x\)-axis) are \((\pm b,0)=(\pm2,0)\).

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, draw a smooth curve connecting these points to form the ellipse. The ellipse will be centered at the origin \((0,0)\) since there are no shifts in the \(x\) or \(y\) terms in the equation.

Graph Description:

The graph is an ellipse centered at the origin. It has vertices at \((0, 3)\) and \((0, - 3)\), and co - vertices at \((2,0)\) and \((-2,0)\). To sketch it, plot these four points and draw a smooth, closed curve passing through them, with the major axis along the \(y\) - axis and the minor axis along the \(x\) - axis.

(Note: Since the question asks to graph the equation, the final answer is the graph of the ellipse as described above. If we were to represent the key points: vertices \((0,3)\), \((0, - 3)\) and co - vertices \((2,0)\), \((-2,0)\) are the main points to plot for the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\))