Unit 5 · Topic 3.5
Sinusoidal Functions
Recognize the sine-and-cosine family and describe its amplitude, midline, period, frequency, symmetry, and changing concavity.
Learning Goals
- Recognize a sinusoidal function from a graph, equation, or description.
- Determine period and frequency and interpret their units.
- Find amplitude and midline from extrema or range.
- Describe changes in concavity across a cycle.
- Use odd and even symmetry to distinguish parent sine and cosine.
1. Define the Sinusoidal Family
A sinusoidal function is obtained from sine or cosine through horizontal and vertical translations and dilations. A general representation is
where \(A\ne0\) and \(B\ne0\). Its graph oscillates smoothly around a horizontal midline with a constant amplitude and period.
2. Sine and Cosine Share One Shape
Cosine is a horizontally shifted sine function:
Therefore, every cosine representation can also be written with sine and vice versa. The choice of function usually depends on which form makes the graph's starting landmark easiest to express.
3. Know the Parent Characteristics
| Feature | \(y=\sin x\) | \(y=\cos x\) |
|---|---|---|
| Amplitude | 1 | 1 |
| Midline | \(y=0\) | \(y=0\) |
| Period | \(2\pi\) | \(2\pi\) |
| Frequency | \(1/(2\pi)\) | \(1/(2\pi)\) |
| Range | \([-1,1]\) | \([-1,1]\) |
| Symmetry | Odd | Even |
4. Read a Sinusoidal Graph
One cycle passes through a repeating sequence of five landmarks: midline, extreme, midline, opposite extreme, and midline. A cosine cycle may instead be anchored at an extreme.
5. Distinguish Period and Frequency
The period \(P\) is input units per cycle. The frequency \(F\) is cycles per input unit:
If a mechanism repeats every 8 seconds, then \(P=8\) seconds per cycle and \(F=1/8\) cycle per second. A longer period means a lower frequency.
6. Calculate Amplitude from Extrema
Amplitude is the nonnegative vertical distance from the midline to either extreme:
If the maximum is 11 and the minimum is 3, the amplitude is
Amplitude is 4, not 8. The difference between maximum and minimum is the total vertical span.
7. Calculate the Midline
The midline is the arithmetic mean of the maximum and minimum:
For extrema 11 and 3, \(D=(11+3)/2=7\). The graph oscillates four units above and below \(y=7\), so its range is \([3,11]\).
8. Move between Range and Characteristics
| Known information | Amplitude | Midline |
|---|---|---|
| Range \([-5,3]\) | \((3-(-5))/2=4\) | \(y=(3+(-5))/2=-1\) |
| Midline \(y=6\), amplitude 2.5 | 2.5 | \(y=6\) |
For the second row, the range is \([6-2.5,6+2.5]=[3.5,8.5]\).
9. Estimate Period from Repeated Landmarks
Measure horizontal distance between consecutive matching landmarks:
Minima or same-direction midline crossings also work. A maximum-to-minimum interval is generally half the period, and adjacent midline crossings are also separated by half a period.
10. Read Concavity without Calculus
A sinusoidal graph repeatedly changes how it bends. It is concave down where tangent slopes decrease as the curve approaches and leaves a maximum, and concave up where slopes increase around a minimum.
For the parent sine graph, concavity changes at its midline crossings. The same alternating bend pattern appears in every cycle.
11. Use Parent-Function Symmetry
The parent sine graph has rotational symmetry about the origin and is odd. The parent cosine graph has reflection symmetry across the y-axis and is even.
12. Separate Periodic from Sinusoidal
| Pattern | Periodic? | Sinusoidal? |
|---|---|---|
| Smooth wave with fixed extrema and cycle length | Yes | Potentially |
| Square wave switching instantly between two levels | Yes | No |
| Damped oscillation with shrinking extrema | Not exactly | No single sinusoidal function |
| Irregular seasonal measurements | Approximately | Requires model validation |
Every sinusoidal function is periodic, but not every periodic relationship is sinusoidal.
13. Complete a Graph-Reading Example
A smooth repeating graph has consecutive maxima at \(x=1\) and \(x=7\), maximum output 9, and minimum output 1.
The graph has amplitude 4, midline \(y=5\), range \([1,9]\), period 6, and frequency \(1/6\) cycle per input unit.
14. AP Reasoning Workflow
- Confirm a smooth repeating sine-or-cosine shape.
- Read maximum and minimum outputs.
- Calculate amplitude and midline.
- Measure one full period between matching landmarks.
- Take the reciprocal for frequency.
- Describe symmetry and concavity using the displayed graph.
State input and output units whenever the graph comes from a context.
15. Common Errors
- Using maximum minus minimum as the amplitude without dividing by 2.
- Calling the maximum output the midline.
- Confusing period with frequency.
- Measuring maximum to minimum as a full period.
- Assuming every repeating graph is sinusoidal.
- Claiming every transformed sine function is odd.
- Reporting a negative amplitude.
Key Takeaways
- Identify amplitude, period, midline, and phase shift from a sinusoidal formula or graph.
- Core relationship: \(y=A\sin(B(x-C))+D,\quad P=\frac{2\pi}{|B|}\)
- Error check: Amplitude is \(|A|\), never a negative number.
- A graph has range \([-4,10]\). Find its amplitude and midline.
- Consecutive minima occur at \(x=-2\) and \(x=3\). Find period and frequency.
- Explain why a repeating triangular wave is not sinusoidal.
- Describe where the parent cosine graph is concave down and concave up over one cycle.
- Explain why translating an odd sine graph upward can produce a function that is no longer odd.