Problem 39 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, and ellipses. \(\frac{x^{2}}{16}+\frac{y^{2}}{1}=1\)

Step-by-Step Solution

Answer

The given equation is already in standard form for an ellipse and can be graphed using its parameters.

Step 1: Identify the Conic Section

Recognize the form of the given equation: \( \frac{x^{2}}{16} + \frac{y^{2}}{1} = 1 \). This equation is of the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), indicating that it represents an ellipse. Compare it with the standard form to confirm it's already in the correct format.

Step 2: Rewrite in Standard Form

Because the given equation is already in the standard form for an ellipse (\( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \)), there is no need to rewrite it. The standard form provides parameters \( a^{2} = 16 \) and \( b^{2} = 1 \).

Step 3: Identify Key Parameters

Calculate the key values needed to graph the ellipse. Here, \( a = \sqrt{16} = 4 \) and \( b = \sqrt{1} = 1 \). These values denote the semi-major and semi-minor axes, respectively, with the x-axis being the semi-major axis due to \( a > b \).

Step 4: Graph the Ellipse

Using the center of the ellipse at the origin (0,0), plot the points at \( (4, 0), (-4, 0), (0, 1), and (0, -1) \). These points correspond to the endpoints of the major and minor axes. Sketch the ellipse by drawing a smooth curve through these points.

Key Concepts

Ellipse

An ellipse is a fascinating shape in geometry that resembles an oval. It is essentially a stretched circle. Ellipses are important in both mathematics and the real world, appearing in things like planetary orbits. They are defined by two main parts; the longer axis, called the "major axis," and the shorter one, known as the "minor axis". These axes intersect at a point called the center of the ellipse.
To understand ellipses better, remember:

• They have two foci or focus points, which are used in their mathematical definition.
• Every point on the ellipse has a consistent total distance to these two foci.
• The semi-major axis is half of the major axis, giving the extent of the ellipse's stretch along its longest part.
• The semi-minor axis is half the minor axis, showing the stretch along its shorter part.

An ellipse's overall form is expressed through a mathematical formula, which varies its appearance by altering the axis lengths.

Standard Form

The standard form of an ellipse is a specific mathematical way to express it with precision. When dealing with ellipses, the standard form plays a crucial role. It is given by the equation: \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\] where

• \(a\) is the semi-major axis.
• \(b\) is the semi-minor axis.

The largest value among \(a\) and \(b\) determines which axis is the major axis. If \(a > b\), the major axis is horizontal, and if \(b > a\), it is vertical. The standard form reveals much about the ellipse, including the position of its axes, center, and orientation. This form is valuable for analyzing and graphing elliptic equations. Recognizing an ellipse in this form allows for quick identification of these important properties.

Graphing Conic Sections

Graphing conic sections, such as ellipses, involves a systematic method. These shapes, including circles, parabolas, ellipses, and hyperbolas, make up a fundamental family in geometry. When graphing an ellipse in its standard form:

• Start by identifying the center, which is usually at the origin \((0,0)\) unless otherwise specified.
• Next, plot the foci, which for ellipses are points along the major axis.Then, use the semi-major and semi-minor axes to locate the vertex points on the graph. These are \((a, 0), (-a, 0), (0, b), and (0, -b)\) for an ellipse centered at the origin.
• Draw a smooth, oval shape connecting these points.

Graphing conic sections helps visualize the properties of ellipses, showing how they stretch and orient in space. It makes the abstract numerical properties more tangible, assisting in deeper understanding and application.