Periodic Functions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Understand what makes a function periodic.
  • Identify the basic graphs of sine, cosine, and tangent functions.
  • Determine amplitude, period, and midline for sinusoidal functions.
  • Recognize transformations of trigonometric graphs.
  • Interpret periodic behavior in real-world contexts.

Key Ideas

A periodic function repeats its values in a regular pattern.

For a periodic function: \[ f(x+P)=f(x) \]

where \(P\) is the period.

The most common periodic functions are:

  • sine: \(\sin(x)\)
  • cosine: \(\cos(x)\)
  • tangent: \(\tan(x)\)

These functions repeat indefinitely.

Sine and Cosine

Basic sine function: \[ y=\sin(x) \]

Basic cosine function: \[ y=\cos(x) \]

Key features:

  • Period: \[ 2\pi \]
  • Range: \[ -1 \le y \le 1 \]
  • Amplitude: \[ 1 \]

Parent graphs of the sine and cosine functions over two periods.

Tangent Function

Basic tangent function: \[ y=\tan(x) \]

Key features:

  • Period: \[ \pi \]
  • Undefined at: \[ x=\frac{\pi}{2}+k\pi \]
  • Has vertical asymptotes.

Basic tangent graph showing repeating branches and vertical asymptotes.

General Sinusoidal Form

A transformed sine or cosine function can be written as: \[ y=a\sin(bx)+d \]

or \[ y=a\cos(bx)+d \]

where:

  • \(|a|\) = amplitude
  • Period: \[ \frac{2\pi}{|b|} \]
  • \(d\) = vertical shift / midline

Transformed sine graph showing amplitude and midline.

Common Problem Types

Identifying Period

For: \[ y=\sin(3x) \]

Period: \[ \frac{2\pi}{3} \]

because: \[ \frac{2\pi}{|3|} = \frac{2\pi}{3} \]

Finding Amplitude

For: \[ y=-4\cos(x) \]

Amplitude: \[ 4 \]

The negative sign reflects the graph but does not affect amplitude.

Identifying Vertical Shift

For: \[ y=\sin(x)+2 \]

The graph is shifted upward by 2 units.

Midline: \[ y=2 \]

Graph Interpretation

ACT problems may ask you to identify:

  • maximum value
  • minimum value
  • period
  • amplitude
  • midline
  • x-values where repeating occurs

Real-World Periodic Modeling

Periodic functions model:

  • sound waves
  • tides
  • seasons
  • Ferris wheel motion
  • daylight hours

Ferris wheel example showing periodic height over time.

Strategies

  • Memorize the parent graph shapes of sine, cosine, and tangent.
  • Amplitude is always the absolute value of the coefficient in front.
  • Period changes when the input is multiplied by a constant.
  • Tangent behaves differently because it has asymptotes and no amplitude.
  • Sketch one full cycle first, then repeat.

Worked Examples

Example 1 — Find the Period

Find the period of: \[ y=\cos(5x) \]

Use: \[ \frac{2\pi}{|b|} \]

Here: \[ b=5 \]

So: \[ \text{Period}= \frac{2\pi}{5} \]

Example 2 — Find Amplitude and Midline

Find the amplitude and midline of: \[ y=3\sin(x)-4 \]

Amplitude: \[ 3 \]

Midline: \[ y=-4 \]

The graph oscillates around \(y=-4\).

Graph of y = 3sin(x) - 4 showing amplitude and midline.

Example 3 — Tangent Period

Find the period of: \[ y=\tan(2x) \]

For tangent: \[ \text{Period}= \frac{\pi}{|b|} \]

Since \(b=2\): \[ \text{Period}= \frac{\pi}{2} \]

Example 4 — Maximum and Minimum Values

Find the maximum and minimum values of: \[ y=2\cos(x)+1 \]

Cosine ranges from: \[ -1 \text{ to } 1 \]

Multiply by 2: \[ -2 \text{ to } 2 \]

Shift up by 1: \[ -1 \text{ to } 3 \]

Minimum: \[ -1 \]

Maximum: \[ 3 \]

WarningCommon Mistakes
  • Forgetting that tangent’s period is \(\pi\), not \(2\pi\).
  • Ignoring absolute value when finding amplitude.
  • Mixing up horizontal and vertical shifts.
  • Forgetting that tangent has asymptotes.
  • Confusing the coefficient outside the function with the coefficient inside the function.

Practice Problems

  1. Find the amplitude and period of: \[ y=5\sin(2x) \]

  2. Find the period of: \[ y=\cos\left(\frac{x}{3}\right) \]

  3. Determine the midline of: \[ y=-2\sin(x)+7 \]

  4. Find the maximum and minimum values of: \[ y=4\cos(x)-1 \]

  5. Find the period of: \[ y=\tan(4x) \]

1.

Amplitude: \[ 5 \]

Period: \[ \frac{2\pi}{2}=\pi \]

2.

Here: \[ b=\frac13 \]

So: \[ \text{Period}= \frac{2\pi}{1/3} = 6\pi \]

3.

Midline: \[ y=7 \]

4.

Cosine ranges from: \[ -1 \text{ to } 1 \]

Multiply by 4: \[ -4 \text{ to } 4 \]

Shift down 1: \[ -5 \text{ to } 3 \]

Minimum: \[ -5 \]

Maximum: \[ 3 \]

5.

For tangent: \[ \text{Period}= \frac{\pi}{4} \]

Summary

  • Periodic functions repeat values in regular intervals.
  • Sine and cosine have period: \[ 2\pi \]
  • Tangent has period: \[ \pi \]
  • Amplitude measures vertical stretch.
  • Midline shows the vertical center of oscillation.
  • Periodic functions are widely used in modeling repeating phenomena.
  • Sine and cosine repeat every \(2\pi\).
  • Tangent repeats every \(\pi\).
  • Amplitude = absolute value of front coefficient.
  • Period changes based on the coefficient inside the function.
  • Tangent has asymptotes; sine and cosine do not.