Problem 34 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. See Using Your Calculator: Graphing Ellipses. \(x^{2}+16 y^{2}=16\)

Step-by-Step Solution

Answer

Graph the ellipse centered at the origin with axes lengths of 8 and 2.

Step 1: Convert Equation to Standard Form of an Ellipse

Given the equation \(x^{2} + 16y^{2} = 16\). To convert this to the standard form of an ellipse, divide each term by 16, getting \(\frac{x^2}{16} + \frac{y^2}{1} = 1\). This is equivalent to \(\frac{x^2}{4^2} + \frac{y^2}{1^2} = 1\), identifying it as an ellipse.

Step 2: Identify Key Features of the Ellipse

The standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) reveals \(a^2 = 16\) and \(b^2 = 1\). Thus, \(a = 4\) and \(b = 1\), meaning the ellipse is centered at the origin \((0,0)\) with the major axis along the x-axis, and the lengths of axes are 8 (major) and 2 (minor).

Step 3: Use Graphing Calculator to Plot

Enter the equation in the graphing calculator as \(\frac{x^2}{16} + \frac{y^2}{1} = 1\). Set the window settings to ensure both axes are visible. Adjust x-axis to at least [-6, 6] and y-axis to [-2, 2] for better visibility.

Step 4: Analyze the Graph

Observe the graph on the calculator display. The plot should represent an ellipse centered at the origin with the longer axis along the x-axis. The points \((\pm 4, 0)\) and \((0, \pm 1)\) should be the ends of the major and minor axes, respectively.

Key Concepts

Standard Form of an Ellipse

An ellipse is a fascinating conic section that extends in two directions. A critical starting point for understanding ellipses is the standard form of their equation, which makes computations simpler and reveals important geometric features. The standard form of an ellipse's equation is given by:

• \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).

This equation describes an ellipse centered at the point \((h, k)\). The terms \(a^2\) and \(b^2\) represent the squares of the semi-major and semi-minor axes lengths, respectively.
For our exercise, we have \(x^2 + 16y^2 = 16\). By dividing each term by 16, we arrive at \(\frac{x^2}{16} + \frac{y^2}{1} = 1\). This tells us it is centered at the origin \((0,0)\) with \(a = 4\) and \(b = 1\). This conversion makes it straightforward to visualize and calculate the dimensions of our ellipse.

Graphing Calculator

A graphing calculator is an invaluable tool in visualizing and understanding mathematical equations, including those of ellipses. By entering the standard form of the ellipse equation into the calculator, you can quickly generate an accurate graph. Here’s how to proceed with our ellipse equation:

• Enter \(\frac{x^2}{16} + \frac{y^2}{1} = 1\) as the equation to be graphed.
• Adjust the window size appropriately to capture the major and minor axes. A suitable window might be: x-axis from \([-6, 6]\), y-axis from \([-2, 2]\).

This allows you to see the complete ellipse on the calculator’s screen. Experiment with different settings to gain a deeper understanding of ellipse graphing.
The graphing calculator helps you see the visual representation instantly, reinforcing your grasp of the equation's implications.

Major and Minor Axes

The major and minor axes are crucial in defining the shape and orientation of an ellipse. Each axis represents the longest and shortest segments that run through the center. In our current context:

• The major axis stretches along the x-direction, with ends at \((\pm 4, 0)\), extending the entire length of 8 units.
• The minor axis is vertical, with ends at \((0, \pm 1)\), totaling 2 units.

To visualize this, imagine stretching an elastic band so that it forms the longest possible horizontal distance, and the shortest vertical distance.
These axes help determine how "stretched" the ellipse is, giving us insights into its geometric proportions and direction.

Conic Sections

Ellipses fall under the fascinating category called conic sections, which are formed by the intersection of a plane and a double-napped cone. Based on the angle and position of intersection, different shapes emerge:

• Circle: A special case of ellipse where both axes are equal.
• Ellipse: Formed when the plane intersects both halves without being parallel to the base.
• Parabola: When the plane is parallel to the generatrix of the cone.
• Hyperbola: When the plane intersects both halves but not parallel to the axis or base.

Understanding conic sections' relationships enriches our comprehension of their behavior in equations. Ellipses specifically show balance between the chaos of a hyperbola and the simplicity of a circle.
Conceptually, they provide insights into orbits and optics, where balance and symmetry are key.