Period, frequency, and phase · Spellman Computer Science

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Key words

Key word · Unit 10PeriodThe shortest positive input shift that repeats the whole pattern.Exactly: A fundamental period $P > 0$ satisfies $f (x + P) = f (x)$ wherever defined, with no smaller positive period.Mix-up alert: For a nonconstant transformed sine or cosine, the period is $\frac{2 π}{| B |}$ ; for tangent it is $\frac{π}{| B |}$ .See this in Key words
Key word · Unit 10Phase shiftA horizontal shift of a repeating graph.Exactly: In A sin(B(x-C))+D or A cos(B(x-C))+D, the phase shift is C, with B nonzero.Mix-up alert: Factor $B$ before reading the shift in $B x - c = B (x - \frac{c}{B})$ . The shift is $\frac{c}{B}$ , with $B \neq 0$ .See this in Key words

1. Period measures repeat distance
For sine and cosine, $y = A \sin (B (x - h)) + D$ or its cosine version has period $P = \frac{2 π}{| B |}$ when A and B are nonzero. The period is measured in input units.
2. Frequency counts cycles
The Key word · Unit 10FrequencyThe number of complete cycles per unit of input.Exactly: For a sinusoid with period $T > 0$ , ordinary frequency is $\frac{1}{T} = \frac{| B |}{2 π}$ cycles per input unit.Mix-up alert: B is angular frequency in radians per input unit, not usually the number of cycles.See this in Key words $f = \frac{1}{P}$ counts cycles per input unit. The angular coefficient $| B | = 2 π f$ counts radians per input unit. A frequency of 3 cycles/s corresponds to $| B | = 6 π$ rad/s.
3. Factor before reading phase
A Key word · Unit 10Phase shiftA horizontal shift of a repeating graph.Exactly: In A sin(B(x-C))+D or A cos(B(x-C))+D, the phase shift is C, with B nonzero.Mix-up alert: Factor $B$ before reading the shift in $B x - c = B (x - \frac{c}{B})$ . The shift is $\frac{c}{B}$ , with $B \neq 0$ .See this in Key words h is visible in $B (x - h)$ . For $\sin (B x - C)$ , factor the input first to find $h = \frac{C}{B}$ ; do not read C alone as the shift.
4. Anchor one complete cycle
A quarter-period is $\frac{P}{4}$ . Start at $x = h$ : a positive-A cosine is at its maximum, while a positive-A sine with $B > 0$ is at the midline rising. Mark five points at $h, h + \frac{P}{4}, h + \frac{P}{2}, h + \frac{3 P}{4}, h + P$ .

Step-through

Follow one checked example

Do this: Move through the four steps and check each number.

Find period, frequency, and phase shift for $y = 3 \sin ((\frac{π}{2}) x - π) + 1$ .

4 steps. Tap a number or Show me how it works.

Worked example

Distinguish two rates

Do this: Follow each step on paper before reading the result.

Step 1: A signal completes 5 cycles in 2 seconds.
Step 2: Its frequency is $\frac{5}{2} = 2.5$ cycles/s.
Step 3: Its period is $\frac{1}{2.5} = 0.4$ s.
Step 4: Its angular coefficient is $2 π (2.5) = 5 π$ rad/s, not 2.5 rad/s.

Worked example

Place a shifted cycle

Do this: Follow each step on paper before reading the result.

Step 1: For $y = 2 \cos (3 (x + \frac{π}{6}))$ , the phase shift is $\frac{π}{6}$ left.
Step 2: $P = \frac{2 π}{3}$ , so $\frac{P}{4} = \frac{π}{6}$ .
Step 3: Starting at $x = \frac{- π}{6}$ , the five outputs are 2,0,-2,0,2.
Step 4: The five $x$ values are $\frac{- π}{6}, 0, \frac{π}{6}, \frac{π}{3}, \frac{π}{2}$ , spanning exactly one full period.

Mix-up alert

Common mistake

Do this: Explain the wrong idea and the repair.

Wrong idea: in $\sin (4 x - 2)$ , the shift is 2 units right. Repair: $4 x - 2 = 4 (x - \frac{1}{2})$ , so the shift is $\frac{1}{2}$ right. Also keep frequency $\frac{1}{P}$ separate from angular coefficient $| B | = \frac{2 π}{P}$ .

Tip

Do Now · Easy · Paper reference

Do this: Use paper. State the definition before you calculate.

Explain why multiplying the input by 2 halves a sinusoid’s period.

Worked example

Guided practice · Medium · Paper reference

Do this: Work on paper and explain each step before checking it.

For $y = \sin (6 x - π)$ :

Step 1: Factor to $6 (x - \frac{π}{6})$ .
Step 2: Shift right $\frac{π}{6}$ .
Step 3: The period is $\frac{π}{3}$ .

Watch

Video: Period, frequency, and phase

Do this: Watch the public lesson, pause to try a step, then return to your practice.

Watch on YouTubeOpens in a new tab. Keep your work here.

Watch for:

Factoring the input to read the phase shift and marking one complete cycle.
Pause before a worked step and try it on paper. Compare the reasoning, not just the final answer.

Open on YouTube

From The Organic Chemistry Tutor · Graphing Trigonometric Functions, Phase Shift, Period, Transformations, Tangent, Cosecant, Cosine

The linked video is free to watch. The creator may offer optional paid resources; they are not needed for this lesson. If YouTube is blocked, use the lesson notes and worked examples here.

Step 1 of 4: Learn it

Build the idea

Key words

1. Period measures repeat distance

2. Frequency counts cycles

3. Factor before reading phase

4. Anchor one complete cycle

Follow one checked example

Distinguish two rates

Place a shifted cycle

Common mistake

Do Now · Easy · Paper reference

Guided practice · Medium · Paper reference

Video: Period, frequency, and phase

Watch for:

Step 2 of 4: Practice it

Recall and explain

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Test the statements

Statement 1 of 4

Practice set 1

Practice set 2

Explain your reasoning

Explain the difference between P, f, and B for a cosine model completing one cycle every 6 seconds.

Extra practice · Hard

A student says the shift of sin⁡(2x−π) is π. Explain the error and give an equivalent model with a different reported shift.

Match the key words

Word match round

Terms

Meanings

Step 3 of 4: Homework

Period, frequency, and phase practice preview

Step 4 of 4: Finish

Ready to move on?

Step 1 of 4: Learn it

Build the idea

Key words

1. Period measures repeat distance

2. Frequency counts cycles

3. Factor before reading phase

4. Anchor one complete cycle

Follow one checked example

Distinguish two rates

Place a shifted cycle

Common mistake

Do Now · Easy · Paper reference

Guided practice · Medium · Paper reference

Video: Period, frequency, and phase

Watch for:

Step 2 of 4: Practice it

Recall and explain

Question

Answer

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A student says the shift of $\sin (2 x - π)$ is $π$ . Explain the error and give an equivalent model with a different reported shift.

A student says the shift of $\sin (2 x - π)$ is $π$ . Explain the error and give an equivalent model with a different reported shift.