Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the curve represented by the parametric equations. Curtate cycloid: \(x=8 \theta-4 \sin \theta, y=8-4 \cos \theta\)

Step-by-Step Solution

Answer

The graph of the parametric equations forms a curtate cycloid, which resembles waves or loops along a horizontal line. Each full cycle of the graph represents one round of parameter \( \theta \) changing from 0 to \( 2\pi \).

Step 1 - Understand the Parametric Equations

The given parametric equations are for a curtate cycloid, which is a form of cycloid. In these equations, \( \theta \) is the parameter that generates the coordinates (x, y) on the graph. For any given value of \( \theta \), these equations will produce a corresponding x and y coordinate.

Step 2 - Prepare to Plot the Graph

A graphing tool that can plot parametric equations is needed. A programmable calculator, an online graphing tool or a mathematical software could be used. Log in and choose the option to graph a parametric curve.

Step 3 - Input the Equations

Enter the given parametric equations into your selected graphing tool. In most graphing utilities, you will need to input x = 8\( \theta \) - 4sin\( \theta \) and y = 8 - 4cos\( \theta \) as separate equations. Make sure to specify that these are parametric equations.

Step 4 - Plot the Graph

Choose an appropriate interval for \( \theta \) to see a good representation of the curve. Usually a single cycle from \( \theta = 0 \) to \( \theta = 2\pi \) is appropriate, but you can adjust the interval as needed. Then, plot the graph by executing or running the command in the graphing tool.

Step 5 - Interpret the Graph

After plotting the graph, notice the shape and pattern produced by the parametric equations. Adjust the range of the x and y axes if necessary to clearly see the entire curve.