Problem 86 - A Graphical Approach to Precalculus with Limits

Question

RELATING CONCEPTS For individual or group investigation (Exercises \(85-90\) ) Consider the ellipse and hyperbola defined by $$ \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 \quad \text { and } \quad \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 $$ respectively. Work Exercises \(85-90\) in order. Graph the ellipse with a calculator, and trace to find the coordinates of several points on the ellipse.

Step-by-Step Solution

Answer

Graph the ellipse using \(\frac{x^2}{16} + \frac{y^2}{12} = 1\) and find points like \((4, 0)\) and \((0, \sqrt{12})\) using a calculator.

Step 1: Identify the Ellipse Equation

The given ellipse is described by the equation \(\frac{x^2}{16} + \frac{y^2}{12} = 1\). This tells us that it is a standard ellipse centered at the origin \((0, 0)\), where the semi-major axis length is 4 along the x-axis and the semi-minor axis length is \(\sqrt{12} \approx 3.46\) along the y-axis.

Step 2: Set Up the Graphing Calculator

Using a graphing calculator, input the ellipse equation \(\frac{x^2}{16} + \frac{y^2}{12} = 1\). Make sure the calculator is set to the correct mode (Rectangular Coordinates) and the viewing window is adjusted to accommodate the ellipse’s major and minor axes ranges, ideally something like x: (-5, 5) and y: (-4, 4).

Step 3: Graph the Ellipse

Press the graph function button on your calculator. The graph should display an elongated circle stretched along the x-axis, representing the ellipse. Check that it is symmetric along both the x-axis and the y-axis.

Step 4: Trace and Record Points

Use the calculator's trace function to explore the ellipse. Move along the curve and stop at several points to record their coordinates. Typically, you should trace and record at least four points: two on the major axis and two on the minor axis. Example points might be \((4, 0)\), \((-4, 0)\), \((0, \sqrt{12})\), \((0, -\sqrt{12})\).

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that resembles a stretched circle. It's characterized by two axes: the major and the minor axis. In the equation provided, \(\frac{x^2}{16} + \frac{y^2}{12} = 1\), the terms indicate how far the ellipse stretches in both directions. Here, the major axis is along the x-axis because the term under \(x^2\) (16) is larger than the one under \(y^2\) (12).

• The full length of the major axis is \(2a\). So, for this equation, \(a = 4\); hence, the total length is 8 units.
• The minor axis length is \(2b\). As \(b^2 = 12\), \(b\) is approximately 3.46, making the total length just under 7 units.

The center of the ellipse is at \((0,0)\), providing symmetry about both axes.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical concepts such as ellipses. It helps you see what equations look like in the geometric space, making it easier to understand relationships and properties. To graph an ellipse, input the equation into your calculator. Adjust the viewing window settings to encapsulate the potential full reach of the ellipse based on its axes:

• Set x-values from \(-5\) to \(5\) so the calculator captures the entire major axis.
• Set y-values from \(-4\) to \(4\) for the minor axis.

This setup ensures that you get a clear visualization of the ellipse on your screen, allowing you to further explore the curve with tracing.

Equations of Conics

Conic sections include ellipses, circles, parabolas, and hyperbolas. Each has its characteristic equation form:

• Circle: \(x^2 + y^2 = r^2\) (all distances from center equal)
• Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (scaled by axes lengths)
• Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) (difference of squared terms)

The given ellipse equation is part of these conic forms. Recognizing this helps streamline the graphing process and highlights different properties such as symmetry, center, and axis alignments inherent in conic sections.

Tracing Points

Tracing points on a graph using a calculator allows you to capture specific coordinates along an ellipse or any curve. This function enables understanding of how the points relate to the characteristics of the conic section. By tracing,

• You can find intercepts where the ellipse meets the axes like \((4, 0)\) and \((0, \sqrt{12})\).
• Tracing gives a concrete understanding of perspectives, such as how the curve's gradient and curvature change.

Using the trace function complements theoretical analysis and facilitates a deeper grasp of the geometry involved in the ellipse, each calculated point representing a tangible aspect of the equation.