Question

graph each equation.

1. \\( \dfrac{x^2}{4} + \dfrac{y^2}{9} = 1 \\)

coordinate plane with x-axis from -8 to 8 and y-axis from -8 to 8, grid lines present

Explanation:

Step1: Identify the conic section type

The equation is \(\frac{x^{2}}{4}+\frac{y^{2}}{9} = 1\), which is in the standard form of an ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (since \(a^{2}=9\) and \(b^{2} = 4\), and \(a>b\), so it is a vertical - major - axis ellipse). The center of the ellipse is at the origin \((0,0)\) because the equation is of the form \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1\) with \(h = 0\) and \(k=0\).

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 \(a=\sqrt{9}=3\), and \(b^{2}=4\), so \(b = \sqrt{4}=2\).

Step3: Determine the vertices and co - vertices

• Vertices: For a vertical - major - axis ellipse centered at the origin, the vertices are at \((0,\pm a)\). Substituting \(a = 3\), the vertices are \((0,3)\) and \((0, - 3)\).
• Co - vertices: For a vertical - major - axis ellipse centered at the origin, the co - vertices are at \((\pm b,0)\). Substituting \(b = 2\), the co - vertices are \((2,0)\) and \((- 2,0)\).

Step4: Plot the points and draw the ellipse

1. Plot the center \((0,0)\).
2. Plot the vertices \((0,3)\) and \((0, - 3)\) (these are the points on the \(y\) - axis, 3 units above and below the center).
3. Plot the co - vertices \((2,0)\) and \((- 2,0)\) (these are the points on the \(x\) - axis, 2 units to the right and left of the center).
4. Then, draw a smooth ellipse passing through these four points. The ellipse will be wider along the \(y\) - axis (since \(a>b\)) and symmetric about both the \(x\) - axis and \(y\) - axis.

To graph the ellipse:

• Mark the center at \((0,0)\).
• Mark the points \((0,3)\), \((0, - 3)\), \((2,0)\), and \((- 2,0)\).
• Connect these points with a smooth, oval - shaped curve. The curve should be symmetric with respect to both the \(x\) - axis and \(y\) - axis.

Answer:

The graph is an ellipse centered at the origin \((0,0)\) with vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\), and it is drawn by plotting the center, vertices, co - vertices and then drawing a smooth curve through them. (The actual graph is an ellipse with the described key points and symmetric about the coordinate axes.)