Problem 10 - A Graphical Approach to College Algebra

Question

If an ellipse has endpoints of the minor axis and vertices at \((-3,0),(3,0),(0,5),\) and \((0,-5)\) what is its domain? What is its range?

Step-by-Step Solution

Answer

Domain: \([-3, 3]\), Range: \([-5, 5]\).

Step 1: Identify the Center

First, we identify the center of the ellipse. Given the vertices \( (0,5) \) and \( (0,-5) \), and the endpoints of the minor axis \( (3,0) \) and \( (-3,0) \), we can determine that the center of the ellipse is at the origin, \( (0,0) \).

Step 2: Determine the Length of Axes

Next, we find the lengths of the major and minor axes. The distance between the vertices \( (0,5) \) and \( (0,-5) \) is \( 10 \), so the length of the major axis is \( 10 \). Consequently, the semi-major axis is \( a = 5 \). The distance between the endpoints of the minor axis \( (-3,0) \) and \( (3,0) \) is \( 6 \), so the length of the minor axis is \( 6 \). Consequently, the semi-minor axis is \( b = 3 \).

Step 3: Understand Orientation

With vertices along the y-axis, this ellipse is vertically oriented, having its major axis along the y-axis and its minor axis along the x-axis.

Step 4: Formulate the Equation

The standard form equation of an ellipse centered at the origin with a vertical major axis is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). Substituting the known values for \( a \) and \( b \), we get the equation \(\frac{x^2}{3^2} + \frac{y^2}{5^2} = 1\), which simplifies to \ \frac{x^2}{9} + \frac{y^2}{25} = 1\.

Step 5: Find the Domain

The domain refers to all possible x-values for the ellipse. Since the length of the minor axis is \( 6 \), the domain is from \(-3\) to \(3\). Hence, the domain is \([-3, 3]\).

Step 6: Find the Range

The range refers to all possible y-values for the ellipse. Since the length of the major axis is \( 10 \), the range extends from \(-5\) to \(5\). Hence, the range is \([-5, 5]\).

Key Concepts

Major Axis

In an ellipse, the major axis is the longest line that can be drawn through the center. This axis defines the longest dimension of the ellipse and runs through both foci.
In the given problem, the major axis lies along the y-axis, connecting the points \((0,5)\) and \((0,-5)\).
This means that the length of the major axis is \(10\).

• The major axis is vertical in orientation.
• It is always the longer of the two axes in an ellipse.

Knowing the major axis allows us to understand the vast extent of the ellipse along one dimension.

Minor Axis

The minor axis of an ellipse is the shortest line that runs through its center. It is perpendicular to the major axis and its endpoints lie at the widest part of the ellipse's oval shape.
For this problem, the minor axis connects the points \((-3,0)\) and \((3,0)\), resulting in a length of \(6\).

• The minor axis is horizontal.
• It is the shorter diameter of the ellipse.

Understanding the minor axis is crucial as it helps define the boundary of the ellipse along its narrower dimension.

Center of Ellipse

The center of an ellipse is the midpoint of both the major and minor axes.
It's where the two axes intersect.
For the ellipse in question, the given points help us locate its center at the origin, \((0,0)\).

• The center is crucial as it serves as the reference point for measuring the lengths and orientations of the axes.
• In standard form, the center coordinates are used to represent the equation of the ellipse.

Understanding the center allows us to contextualize the position and orientation of the ellipse in a coordinate plane.

Domain and Range

In mathematics, determining the domain and range of an ellipse involves identifying the possible x and y values represented by its shape.
For this ellipse:

• The **domain** is the set of all x-values: due to the minor axis being \(6\) units wide, the domain is \([-3, 3]\).
• The **range** is the set of all y-values: because the major axis is \(10\) units long, the range is \([-5, 5]\).

By calculating domain and range, students can describe the horizontal and vertical span of the ellipse within the coordinate system.