Problem 33 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. See Using Your Calculator: Graphing Ellipses. \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)

Step-by-Step Solution

Answer

Graph the ellipse centered at the origin with a major axis of 6 along the x-axis and a minor axis of 4 along the y-axis using a graphing calculator.

Step 1: Identify the Equation Form

Recognize that the equation \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \) represents an ellipse in standard form. The standard form is \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), where \( a^2 = 9 \) and \( b^2 = 4 \). Thus, \( a = 3 \) and \( b = 2 \).

Step 2: Determine the Center and Orientation

Since the equation is centered at the origin and both \( a \) and \( b \) are associated with \( x \) and \( y \) respectively, the ellipse is oriented along the x and y axes. The center is at (0,0).

Step 3: Calculate the Major and Minor Axis Lengths

For this ellipse, \( a = 3 \) and represents the semi-major axis length along the x-axis. \( b = 2 \), representing the semi-minor axis length along the y-axis. The lengths of the axes are 6 (major) and 4 (minor).

Step 4: Graph Using a Calculator

Access your graphing calculator and input the equation \( \sqrt{9 - 9\frac{y^{2}}{4}} \) for the top half of the ellipse and \( -\sqrt{9 - 9\frac{y^{2}}{4}} \) for the bottom half. Additionally, for the other orientation, plot \( \sqrt{4 - \frac{4}{9}x^{2}} \) and \( -\sqrt{4 - \frac{4}{9}x^{2}} \). Ensure each equation is set separately to maintain a clear graph.

Step 5: Interpret the Graph

Once the graphs are plotted, you should see an ellipse centered at the origin. The ellipse should extend 3 units from the origin along the x-axis and 2 units along the y-axis. Check that the shape appears symmetric around both axes.

Key Concepts

Standard Form of Ellipse

Ellipses have a unique characteristic in the realm of conic sections. They hug the coordinate plane tightly within their bounds. To express an ellipse mathematically, we use its standard form equation: \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\] In this formula:

• \(a\) represents the horizontal radius (along the x-axis) and is always placed beneath \(x^{2}\).
• \(b\) represents the vertical radius (along the y-axis) and is always placed beneath \(y^{2}\).

This arrangement helps identify how far the ellipse extends in either direction. The numbers underneath \(x^{2}\) and \(y^{2}\) in our equation determine the shape and orientation of the ellipse within the coordinate plane. In our equation \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\), \(a^{2}=9\) and \(b^{2}=4\), which simplify to \(a=3\) and \(b=2\). These values directly influence the size and stretch of the ellipse.

Major and Minor Axes

In every ellipse, there are two crucial lines called the major and minor axes. The major axis is the longest diameter, stretching from one end of the ellipse to the other. It's the line where our values for \(a\) or \(b\) come into play. Here’s how it works for our current ellipse:

• The major axis corresponds to \(a\), the larger of the two values. In our case, \(a=3\), so the major axis runs along the x-axis.
• This axis is 6 units long, doubling the \(a\) value since it extends equally in both directions from the center point.
• The minor axis is defined by \(b\). For this ellipse, \(b=2\), making it the smaller axis and running vertically along the y-axis.
• The minor axis measures 4 units in total length, again by doubling \(b\).

The intersection of these axes marks the ellipse’s center, which is positioned at the origin, \((0, 0)\), for our example.

Graphing Calculator

Using a graphing calculator is essential for accurately visualizing conic sections like ellipses. Here, the calculator helps you plot complex equations by breaking them down into manageable parts. To graph an ellipse, you perform the following steps:

• Enter the equation in segments. For the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\), you can split it into two pairs of functions. One pair represents the top and bottom halves, another the left and right halves.
• Convert to a format the calculator understands. For our ellipse, input \(\sqrt{9 - 9\frac{y^{2}}{4}}\) and its negative counterpart for the vertical halves. For horizontal halves, use \(\sqrt{4 - \frac{4}{9}x^{2}}\) and its negative version.
• Ensure clear visualization. Input these functions separately to ensure they map correctly on the screen.

Once you input these equations, the graphing calculator displays the ellipse clearly, allowing you to verify features such as symmetry and axis lengths. This visualization reinforces your understanding of how ellipses function spatially within the coordinate system.