Problem 4 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The ellipse is centered at (0, 0), major axis endpoints: (0, 2) & (0, -2), minor axis endpoints: (1, 0) & (-1, 0), foci: (0, √3) & (0, -√3).

Step 1: Recognize the Standard Form of the Ellipse

The given equation is \( x^2 + \frac{y^2}{4} = 1 \). This equation resembles the standard form of an ellipse \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) with center at \((h, k)\), which simplifies here to \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \). Here, the center is at \((h, k) = (0, 0)\), \(a^2 = 1\), and \(b^2 = 4\).

Step 2: Determine the Lengths of the Axes

From the equation \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \), we have \( a^2 = 1 \) and \( b^2 = 4 \). Thus, \( a = \sqrt{1} = 1 \) and \( b = \sqrt{4} = 2 \). The major axis is vertical since \( b > a \) and the semi-major axis length is \( b = 2 \). The semi-minor axis is \( a = 1 \).

Step 3: Identify the Endpoints of Each Axis

Since the ellipse is centered at the origin \((0, 0)\), the endpoints of the vertical major axis are \((0, b)\) and \((0, -b)\) which are \((0, 2)\) and \((0, -2)\). The endpoints of the horizontal minor axis are \((a, 0)\) and \((-a, 0)\), which are \((1, 0)\) and \((-1, 0)\).

Step 4: Calculate and Label the Foci

The distance of the foci from the center along the major axis is calculated using \( c = \sqrt{b^2 - a^2} = \sqrt{4 - 1} = \sqrt{3} \). Thus, the foci are located at \((0, \sqrt{3})\) and \((0, -\sqrt{3})\).

Step 5: Sketch the Ellipse and Label Key Points

Draw the ellipse centered at the origin, elongated vertically due to the major axis. Label the endpoints of the major axis at \((0, 2)\) and \((0, -2)\), and the endpoints of the minor axis at \((1, 0)\) and \((-1, 0)\). Mark the foci on the vertical major axis at \((0, \sqrt{3})\) and \((0, -\sqrt{3})\).

Key Concepts

Graphing Ellipses

When graphing ellipses, we begin by understanding the shape and position based on its equation. Ellipses have a unique property—they are oval-shaped curves with two symmetry axes. Let's break down how to sketch the graph effectively:

• Identify the center: The center is where the two axes of symmetry intersect. In this exercise, the center is at the origin, (0, 0).
• Determine the lengths of the semi-major and semi-minor axes: For the given ellipse, we found that the vertical axis is longer (2 units), and the horizontal axis is shorter (1 unit).
• Plot the endpoints: Mark the end points on both axes. For the vertical major axis, these are (0, 2) and (0, -2). For the horizontal minor axis, these are (1, 0) and (-1, 0).

To complete the graph, simply draw a smooth curve connecting these points, forming the structured shape of an ellipse. Paying attention to symmetry will help you achieve an accurate representation.

Foci of an Ellipse

The foci of an ellipse are two points lying along the major axis, crucial in defining its shape. To find these foci:

• Use the formula: The position of the foci depends on the length of the semi-major (b) and semi-minor (a) axes, calculated with \( c = \sqrt{b^2 - a^2} \).
• Calculate for our ellipse: With b = 2 and a = 1, the distance to the foci from the center, c, is found using the formula above. This results in \( c = \sqrt{4 - 1} = \sqrt{3} \).
• Position the foci: Since the major axis is vertical, the foci's coordinates are (0, \( \sqrt{3} \)) and (0, -\( \sqrt{3} \)).

Understanding and marking the foci help in drawing an accurate ellipse, as every point on the ellipse is such that the sum of the distances from the foci is constant.

Standard Form of an Ellipse

The standard form of an ellipse equation provides essential information about its properties. The standard form is stated as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where (h, k) is the center.

• Identify the structure: In our given equation, \( x^2 + \frac{y^2}{4} = 1 \), it modifies to \( \frac{x^2}{1} + \frac{y^2}{4} = 1 \).
• Find the center: The values of h and k are both zero, positioning the center at (0, 0).
• Determine a and b: Here, a^2 is 1 and b^2 is 4, giving us a = 1 and b = 2.

This form highlights whether the major axis is horizontal or vertical, depending on the comparative sizes of a and b. It's a fundamental part of solving and graphing ellipse equations.