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 \(b^{2} = 4\), and \(a>b\), so it is a vertical ellipse). For an ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), the center is at \((0,0)\), the vertices are at \((0,\pm a)\) and the co - vertices are at \((\pm b,0)\).

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

Given \(a^{2}=9\), then \(a = 3\) (we take the positive value since it represents a length). Given \(b^{2}=4\), then \(b = 2\) (we take the positive value since it represents a length).

Step3: Determine the key points

• Vertices: Since it is a vertical ellipse, the vertices are at \((0,a)=(0,3)\) and \((0, - a)=(0,-3)\).
• Co - vertices: The co - vertices are at \((b,0)=(2,0)\) and \((-b,0)=(- 2,0)\).

Step4: Graph the ellipse

Plot the center \((0,0)\), 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. The major axis is along the \(y\) - axis with length \(2a = 6\) and the minor axis is along the \(x\) - axis with length \(2b=4\).

To graph the ellipse:

1. Mark the center at the origin \((0,0)\).
2. Move 3 units up and down from the center to mark the vertices \((0,3)\) and \((0, - 3)\).
3. Move 2 units left and right from the center to mark the co - vertices \((-2,0)\) and \((2,0)\).
4. Draw a smooth elliptical curve passing through these four points. The ellipse will be taller than it is wide, symmetric about both the \(x\) - axis and \(y\) - axis.

(Note: Since the question is to graph the equation, the final answer is the graph of the ellipse with the above - mentioned key points. If we were to describe the graph in words, it is an ellipse centered at the origin, with vertices at \((0,\pm3)\) and co - vertices at \((\pm2,0)\))

Answer:

The graph is an ellipse centered at \((0,0)\), with vertices at \((0, 3)\), \((0, - 3)\) and co - vertices at \((2, 0)\), \((-2, 0)\), and a smooth curve connecting these points (symmetric about both axes, taller along the \(y\) - axis).