Exercises: 4.3 Periodic Functions
Practice each skill in the Homework Problems listed:
- Graph periodic functions
- Write equations for sinusoidal functions
- Graph sinusoidal functions
- Find amplitude, period, and midline
- Fit a sinusoidal function to data or to a description
- Find coordinates of points on a sinusoidal graph
- Identify periodic functions and give their periods
- Sketch graphs to model sinusoidal functions
- Analyze periodic graphs
Problems: #14, 8, 10, 12, 16, 18, 20, 30, 38, 40, 46, 48, 50, 52, 56, 60, 70
1.
An ant is walking clockwise around the face of a sundial in the garden. The sundial is a circle with a 12-inch diameter, and the ant makes one circuit of the sundial in 24 seconds.
- Sketch the sundial and a coordinate system with its origin at the center of the dial. Draw the [latex]y[/latex]-axis to align with the gnomon (pointer) of the sundial.
- Suppose we start timing the ant when it is at the tip of the gnomon. Complete the table showing the ant’s location as an angle in standard position, and its [latex]y[/latex]-coordinate at that time.
| [latex]t[/latex] |
0 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
22 |
24 |
| [latex]\theta[/latex] |
[latex]90^{o}[/latex] |
[latex]60^{o}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]y = f(t)[/latex] |
[latex]6[/latex] |
[latex]3\sqrt{3}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch a graph of the ant’s [latex]y[/latex]-coordinate as a function of time.
- If the ant makes a second trip around the sundial, what will the graph look like from [latex]t = 24[/latex] to [latex]t = 48[/latex] ? Explain the statement [latex]f(t + 24) = f(t).[/latex]
2.
Repeat Problem 1, but make a table and graph of the ant’s [latex]x[/latex]-coordinate as a function of time.
| [latex]t[/latex] |
0 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
22 |
24 |
| [latex]\theta[/latex] |
[latex]90^{o}[/latex] |
[latex]60^{o}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]x = g(t)[/latex] |
[latex]0[/latex] |
[latex]