Question

graph each equation.

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

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

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 - major - axis ellipse).

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

For the ellipse equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), we have \(a^{2}=9\), so \(a = 3\) (the distance from the center to the vertices along the \(y\) - axis), and \(b^{2}=4\), so \(b = 2\) (the distance from the center to the co - vertices along the \(x\) - axis). The center of the ellipse \((h,k)\) is \((0,0)\) (since the equation is \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1=\frac{(x - 0)^{2}}{2^{2}}+\frac{(y - 0)^{2}}{3^{2}}\)).

Step3: Find the vertices and co - vertices

• Vertices: Since the major axis is along the \(y\) - axis, the vertices are \((0,\pm a)=(0,\pm3)\).
• Co - vertices: Since the minor axis is along the \(x\) - axis, the co - vertices are \((\pm b,0)=(\pm2,0)\).

Step4: Plot the points and draw the ellipse

1. Plot the center \((0,0)\).
2. Plot the vertices \((0,3)\) and \((0, - 3)\).
3. Plot the co - vertices \((2,0)\) and \((-2,0)\).
4. Then, sketch the ellipse by connecting these points smoothly. The ellipse will be symmetric about both the \(x\) - axis and \(y\) - axis.

To graph the ellipse:

• Mark the center at the origin \((0,0)\).
• Move 3 units up and down from the center to get the vertices \((0,3)\) and \((0, - 3)\).
• Move 2 units left and right from the center to get the co - vertices \((-2,0)\) and \((2,0)\).
• Draw a smooth curve connecting these four points (and the rest of the ellipse) such that the curve is symmetric about both the \(x\) - axis and \(y\) - axis.

Answer:

The graph is an ellipse with center \((0,0)\), vertices \((0,\pm3)\), co - vertices \((\pm2,0)\), and is symmetric about the \(x\) - axis and \(y\) - axis. (The actual drawing would show a vertical - major - axis ellipse passing through \((\pm2,0)\) and \((0,\pm3)\) with a smooth curve.)