Problem 29 - Algebra and Trigonometry

Question

Use a graphing device to graph the ellipse. $$ \frac{x^{2}}{25}+\frac{y^{2}}{20}=1 $$

Step-by-Step Solution

Answer

Graph the ellipse centered at (0,0), with vertices (5,0) and (-5,0), and co-vertices approximately at (0, 4.47) and (0, -4.47).

Step 1: Identify the Standard Form of the Ellipse

The equation given, \( \frac{x^2}{25} + \frac{y^2}{20} = 1 \), is in the standard form of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), with \( a^2 = 25 \) and \( b^2 = 20 \). This is a horizontal ellipse because \( a^2 \) is greater than \( b^2 \).

Step 2: Find the Lengths of the Semi-Axes

The lengths of the semi-major and semi-minor axes are determined by \( a \) and \( b \). Calculate \( a \) and \( b \) as follows: \( a = \sqrt{25} = 5 \) and \( b = \sqrt{20} \approx 4.47 \). So, the semi-major axis is 5 and the semi-minor axis is approximately 4.47.

Step 3: Plot the Center of the Ellipse

The standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) indicates that the ellipse is centered at the origin \((0,0)\). Plot this center point on the graph.

Step 4: Draw the Axes and Plot the Vertices

Since the ellipse is horizontal, the major axis is along the x-axis. Plot the vertices at \((0 \pm a, 0)\), which are \((5, 0)\) and \((-5, 0)\). The co-vertices are at \((0, 0 \pm b)\), which are approximately \((0, 4.47)\) and \((0, -4.47)\).

Step 5: Sketch the Ellipse

Using the vertices and the co-vertices as guides, sketch the ellipse, ensuring that it passes through these points. The ellipse should be elongated along the x-axis.

Key Concepts

Ellipse Standard Form

To understand how to graph an ellipse, it's essential to start with its standard form. For an ellipse centered at the origin, the equation is

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

This formula is crucial because it tells us how the ellipse is oriented and provides the key information we need for graphing.
Notice the denominators: \(a^2\) and \(b^2\). These values help us understand the length and orientation of the ellipse's axes.
In the example equation \(\frac{x^2}{25} + \frac{y^2}{20} = 1\), \(a^2 = 25\) and \(b^2 = 20\). Since \(a^2\) is greater than \(b^2\), this ellipse is horizontal, meaning its longer axis runs parallel to the x-axis.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are key features of an ellipse. They help define the size and shape of the ellipse and determine how it is stretched out.

• The semi-major axis is the longest radius of the ellipse, running from the center to the furthest point on the boundary along the x-axis in a horizontal ellipse.
• The semi-minor axis is the shortest radius of the ellipse, extending from the center to the boundary along the y-axis in a horizontal ellipse.

To find these axes, we take the square root of the denominators from the standard form:

• \(a = \sqrt{25} = 5\)
• \(b = \sqrt{20} \approx 4.47\)

Therefore, for this ellipse:

• the semi-major axis is 5 units
• the semi-minor axis is approximately 4.47 units

Knowing these values gives you the lengths to plot when drawing your ellipse.

Graphing Ellipses Step-by-Step

Graphing ellipses involves careful steps to ensure that the shape is accurately represented. Here's a simple way to follow along:1. **Identify the Center:** Start with the center of the ellipse, which is given by the standard form equation. For our example, it's (0,0).
2. **Determine the Semi-Axes:** Calculate the lengths of the semi-major axis \(a\) and the semi-minor axis \(b\). For our ellipse, \(a = 5\) and \(b \approx 4.47\).
3. **Plot the Center:** On your graphing device, mark the center at (0,0).
4. **Plot the Vertices and Co-Vertices:** Since this is a horizontal ellipse, plot the vertices along the x-axis at \((\pm a, 0)\) or \((5, 0)\) and \((-5, 0)\). The co-vertices will be along the y-axis at \((0, \pm b)\) or approximately \((0, 4.47)\) and \((0, -4.47)\).
5. **Sketch the Ellipse:** Using the vertices and co-vertices as guides, sketch a smooth, oval shape. Ensure it passes through these plotted points and maintains the orientation you calculated.
Following these steps ensures a well-proportioned and accurate graph of the ellipse.

Center of Ellipse

The center of an ellipse is a critical reference point for graphing and understanding the geometry of the shape.

• In the standard form equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the ellipse is centered at the origin (0,0).

This central point is where the axes of the ellipse intersect, and it's the anchor for measuring out the semi-major and semi-minor axes.
Knowing the center helps you accurately draw the ellipse by:

• Guiding the placement of the vertices and co-vertices.
• Ensuring that the ellipse is symmetrical about both axes.

For our example ellipse, the center is at (0,0), simplifying the graphing process since you can easily plot the ellipse around this point.