Question

graph each equation.

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

graph with x-axis from -8 to 8 and y-axis from -8 to 8, grid lines, origin at (0,0)

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 \(b^2=4\), so \(a = 3\), \(b = 2\)) with a vertical major axis (because \(a>b\) and the \(y^2\) term has the larger denominator).

Step2: Find the vertices and co - vertices

• For the \(y\) - axis (major axis) vertices: When \(x = 0\), we solve for \(y\). Substitute \(x = 0\) into the equation \(\frac{0^2}{4}+\frac{y^2}{9}=1\), which simplifies to \(\frac{y^2}{9}=1\), then \(y^2=9\), so \(y=\pm3\). So the vertices are \((0, 3)\) and \((0, - 3)\).
• For the \(x\) - axis (minor axis) co - vertices: When \(y = 0\), we solve for \(x\). Substitute \(y = 0\) into the equation \(\frac{x^2}{4}+\frac{0^2}{9}=1\), which simplifies to \(\frac{x^2}{4}=1\), then \(x^2 = 4\), so \(x=\pm2\). So the co - vertices are \((2, 0)\) and \((- 2, 0)\).

Step3: 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 an ellipse that passes through these points, with the major axis along the \(y\) - axis and the minor axis along the \(x\) - axis.

(Note: Since the question is about graphing, the final answer is the graph of the ellipse with vertices \((0,\pm3)\) and co - vertices \((\pm2,0)\) as described above. If we were to describe the key points for the graph: the ellipse is centered at the origin \((0,0)\), has a vertical major axis with length \(2a = 6\) (from \(y=-3\) to \(y = 3\)) and a horizontal minor axis with length \(2b=4\) (from \(x=-2\) to \(x = 2\)).)

Answer:

The graph is an ellipse centered at the origin \((0,0)\), with vertices at \((0, 3)\) and \((0, - 3)\), and co - vertices at \((2, 0)\) and \((-2, 0)\). 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.