Problem 36 - Precalculus: Mathematics for Calculus

Question

Graphing Ellipses Use a graphing device to graph the ellipse. $$x^{2}+\frac{y^{2}}{12}=1$$

Step-by-Step Solution

Answer

Graph the ellipse centered at the origin with vertical semi-major axis length \(2\sqrt{3}\) and horizontal semi-minor axis length \(1\).

Step 1: Identify the Standard Form of an Ellipse

The general equation for an ellipse is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h,k)\) is the center of the ellipse, and \(a\) and \(b\) are the semi-major and semi-minor axes. In this equation, \(h = 0\) and \(k = 0\), so the center of the ellipse is at the origin \((0,0)\).

Step 2: Compare and Restructure the Given Equation

The given equation is \(x^2 + \frac{y^2}{12} = 1\). Notice that this equation is of the form \(\frac{x^2}{1} + \frac{y^2}{12} = 1\). Here \(a^2 = 1\) and \(b^2 = 12\).

Step 3: Determine the Lengths of the Axes

From \(a^2 = 1\), take the square root to find \(a = 1\). From \(b^2 = 12\), take the square root to find \(b = \sqrt{12} = 2\sqrt{3}\). Thus, the semi-major axis has length \(2\sqrt{3}\) and the semi-minor axis has length \(1\).

Step 4: Identify the Directions of the Axes

Since \(a^2 < b^2\), the semi-major axis is vertical, parallel to the \(y\)-axis. The semi-minor axis is horizontal, parallel to the \(x\)-axis.

Step 5: Graph the Ellipse

The center of the ellipse is \((0,0)\). Plot the endpoints of the semi-major axis at \((0, 2\sqrt{3})\) and \((0, -2\sqrt{3})\). Plot the endpoints of the semi-minor axis at \((1, 0)\) and \((-1, 0)\). Draw a smooth, symmetrical curve connecting these points to form the ellipse.

Key Concepts

Ellipse Standard Form

Ellipses are fascinating shapes that occur frequently in mathematics and nature, making them a vital concept to understand. The standard form of an ellipse provides a clear structure to recognize and analyze these curves. The general equation for an ellipse is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). This format reveals the core components of the ellipse.

• \((h, k)\) represents the center of the ellipse.
• \(a\) is the length of the semi-major axis, while \(b\) is that of the semi-minor axis.

In simple terms, the semi-major axis is the longest radius, and the semi-minor axis is the shortest. Depending on the values of \(a\) and \(b\), you can determine the shape of the ellipse, whether it's stretched more horizontally or vertically. Recognizing this form is crucial for solving problems related to graphing ellipses.

Semi-Major Axis

The semi-major axis of an ellipse is an essential part that decides the overall shape of the ellipse. In the equation of an ellipse, \(a\) typically represents the semi-major axis, although this depends on the positioning and orientation of the ellipse compared to the coordinate axes.

• If \(a > b\), the ellipse is elongated horizontally.
• If \(b > a\), the ellipse is elongated vertically.

In our exercise, we found that \(b^2 = 12\) and \(a^2 = 1\), which means \(b = 2\sqrt{3}\) and \(a = 1\). Since \(b > a\), the semi-major axis is vertical, aligning with the \(y\)-axis. To graph it, locate the center at the origin, then mark points at a distance of \(2\sqrt{3}\) units up and down along the \(y\)-axis. These endpoints are crucial for sketching the ellipse accurately.

Semi-Minor Axis

Complementing the semi-major axis, the semi-minor axis provides the resistance, or the shortest span, of the ellipse. This axis is equally significant in determining the ellipse's full structure. In standard form, \(b\) usually indicates the semi-minor axis when \(b < a\).
The semi-minor axis defines the smaller radius of the ellipse. Its endpoints are situated parallel to the diameter of the semi-major axis. In the given problem, with \(a^2 = 1\) and \(b^2 = 12\), we had \(a = 1\) which indicates the horizontal semi-minor axis. Thus, this axis extends one unit to the left and right from the center. This setup complements the semi-major axis vertically, adding dimensions to the ellipse's precise shape.

Graphing Techniques

Graphing an ellipse correctly combines understanding its algebraic forms with visual plotting techniques. The steps to graphing an ellipse involve both calculation and artistry to ensure an accurate representation.

• Identify the center, typically given by \((h, k)\).
• Calculate the \(a\) and \(b\) values to determine the axes lengths.
• Place the endpoints of the semi-major and semi-minor axes from the center point.
• Use a smooth, symmetrical curve to connect these endpoints.

Ensure that this curve softly bends to connect the four key points (endpoints of the axes) with symmetry around the center. In our specific exercise, the ellipse was centered at \((0,0)\), vertical axis length was \(2\sqrt{3}\), and the horizontal was 2, with vertices marked along these axes. Always double-check that your graph maintains the orientation of the semi-major axis and captures the shape accurately.