Problem 1 - College Algebra with Modeling and Visualization

Question

Graph the ellipse. Label the foci and the endpoints of each axis. $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at the origin, with foci at \((0, \sqrt{5})\) and \((0, -\sqrt{5})\), vertices at \((0, 3)\) and \((0, -3)\), and endpoints of the minor axis at \((2, 0)\) and \((-2, 0)\).

Step 1: Identify the standard form

The given equation is \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \). This equation represents an ellipse in the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a^2 = 4 \) and \( b^2 = 9 \). Clearly, \( a = 2 \) and \( b = 3 \). Since \( b > a \), the major axis is along the y-axis.

Step 2: Determine vertices and endpoints

The vertices of the ellipse lie on the y-axis at points \( (0, b) \) and \( (0, -b) \). Therefore, the endpoints of the major axis are \( (0, 3) \) and \( (0, -3) \). The endpoints of the minor axis are \( (a, 0) \) and \( (-a, 0) \). Thus, these are \( (2, 0) \) and \( (-2, 0) \).

Step 3: Calculate the foci

The foci of the ellipse are determined using the formula \( c = \sqrt{b^2 - a^2} \). Substituting the known values: \( c = \sqrt{9 - 4} = \sqrt{5} \). The foci lie on the major axis (y-axis), so their coordinates are \( (0, \sqrt{5}) \) and \( (0, -\sqrt{5}) \).

Step 4: Sketch the graph

Draw the Cartesian plane. Plot the endpoints of the major axis at \( (0, 3) \) and \( (0, -3) \), the endpoints of the minor axis at \( (2, 0) \) and \( (-2, 0) \). Mark the foci at \( (0, \sqrt{5}) \) and \( (0, -\sqrt{5}) \). Connect these points to draw the ellipse, making sure the ellipse is elongated along the y-axis.

Key Concepts

Standard form of an Ellipse

When working with ellipses, understanding their standard form is critical. The standard form of an ellipse is written as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]This formula helps distinguish the characteristics of the ellipse. In this format, \(a^2\) and \(b^2\) are denominators representing the squares of the respective axis lengths. Whether or not the ellipse is vertical or horizontal depends on the values of \(a\) and \(b\).

• If \(a^2 > b^2\), the ellipse is stretched more in the x-direction, making it horizontal.
• If \(b^2 > a^2\), the ellipse is stretched more in the y-direction, making it vertical.

This form is essential for accurately graphing and identifying the key properties of the ellipse.

Major and Minor Axes

In an ellipse, understanding the major and minor axes is crucial. These axes serve as the reference points around which the ellipse is shaped.

• Major Axis: The longest axis of the ellipse. It determines the dominant direction in which the ellipse is stretched. In our exercise, since \(b^2\) (9) is greater than \(a^2\) (4), the major axis is vertical, lying along the y-axis.
• Minor Axis: The shortest axis of the ellipse. In this case, it's horizontal along the x-axis. It represents the smaller stretch of the ellipse.

The endpoints of these axes, known as vertices, help in defining the precise shape and orientation of the ellipse in the graph.
The confirmation of these axes can be critical for ensuring the accuracy of plotting an ellipse on a coordinate plane.

Vertices and Foci

Vertices and foci are integral parts of an ellipse. They provide pivotal points that define the shape and size of the ellipse.Vertices: These are the points where the ellipse is the widest or tallest. There are a total of four vertices:

• The vertices on the major axis are reached at the topmost and bottommost points, calculated by \((0, b)\) and \((0, -b)\) respectively. In this exercise, they are \((0, 3)\) and \((0, -3)\).
• The vertices on the minor axis are the side limits, expressed as \((a, 0)\) and \((-a, 0)\), thus \((2, 0)\) and \((-2, 0)\).

Foci: The foci of an ellipse are special points on the major axis. They help in maintaining the constant sum of the distances to any point on the ellipse. The formula to find them is: \[ c = \sqrt{b^2 - a^2} \]Here, \(c\) is calculated as \(\sqrt{5}\), placing the foci at \((0, \sqrt{5})\) and \((0, -\sqrt{5})\). These points, along with the vertices, are fundamental in depicting the complete shape of an ellipse.