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D. Trigonometric Functions · Sinusoidal Functions

Sinusoidal Functions: Unit Review

D2.1 D2.2 D2.3 D2.4 D2.5 D2.6 D2.7 D2.8 D3.1 D3.2 D3.3 D3.4 D3.5About 40 minutes
Learning goals
  • I can describe periodic and sinusoidal functions using period, amplitude, axis and range.
  • I can sketch y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c and state their properties.
  • I can write the equation of a sinusoidal function from a graph, properties or data.
  • I can model real situations with sinusoidal functions and solve problems with the models.

Words to know

periodic function
A function whose values repeat at regular intervals: f(x + p) = f(x).
amplitude
(max − min) ÷ 2 = |a|
period
The length of one cycle: 360° ÷ |k|
phase shift
The horizontal translation d, read after factoring k(x − d).
axis of the curve
y = (max + min) ÷ 2 = c

Let's review!

In this unit you met periodic functions and learned the language of cycles: period, amplitude, axis and range. You extended the sine and cosine ratios into the sine and cosine functions, learned the graphs of y = sin x and y = cos x, and studied how the four parameters of y = a sin(k(x − d)) + c stretch, reflect and move them. Then you put the functions to work: writing equations from graphs, fitting models to data, modelling Ferris wheels, tides and daylight, predicting the effect of changes, and solving problems.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. For every equation, say the four properties out loud first: amplitude, period, phase shift, axis. Make a quick sketch for every question about times or intervals. Keep your calculator in degree mode (sin 30 should give 0.5). For each question you get wrong, write down why (factoring? period formula? second angle? units?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

avertical stretch by |a|a < 0: reflect in x-axisamplitude = |a|khorizontal stretch by 1/|k|period = 360° ÷ |k|k = 360 ÷ perioddhorizontal translationphase shift d (factor first!)x − 30°: 30° to the rightcvertical translationaxis y = crange c − |a| to c + |a|y = a sin(k(x − d)) + cory = a cos(k(x − d)) + cmapping rule: (x, y) → (x/k + d, ay + c)
The four parameters on one page. Factor the inside as k(x − d) before reading the phase shift.
−45°0°45°90°135°180°225°−5−4−3−2−10123(30°, 2)(120°, −4)(210°, 2)
Use this graph for the questions in the section "Writing the Equation of a Sinusoidal Function".
Just for fun

This unit in one sentence: everything that goes around comes around, and now you can predict when.

Periodic Functions: Cycle, Period, Amplitude and Axis

1

A periodic function has a maximum of 14 and a minimum of 2. Find the amplitude and the equation of the axis.

2

One cycle of a graph starts at t = 2 s and the next cycle starts at t = 9 s. What is the period?

3

Is the number of hours of darkness each day, over several years, periodic? Is it sinusoidal?

Predicting the Future: Extrapolating Periodic Behaviour

1

A function has period 24 and f(5) = 10. Find f(77).

2

Daylight on day 172 is about 15.4 h. Predict the daylight on day 902.

3

A wheel turns once every 45 s. Which time in the first turn has the same height as t = 1000 s?

From the Sine Ratio to the Sine Function

1

Find the exact value of sin 300°.

2

Find the exact value of sin 480°.

3

Explain why y = sin x is a function but x² + y² = 1 is not.

The Graphs of y = sin x and y = cos x

1

List the zeros of y = cos x for −360° ≤ x ≤ 360°.

2

On which interval is y = cos x increasing for 0° ≤ x ≤ 360°?

3

State the range and period of y = sin x.

Transformations I: Vertical Stretches, Reflections and Translations (a and c)

1

State the range of y = −4 sin x + 1.

2

Where does the point (270°, −1) on y = sin x go under y = 2 sin x + 3?

3

Write a cosine function (a > 0, no horizontal change) with maximum 9 and minimum 1.

Transformations II: Horizontal Stretches and the Period (k)

1

State the period of y = cos 6x.

2

Find k (k > 0) for a period of 40°.

3

List the key points of one cycle of y = sin 4x.

Transformations III: Horizontal Translations and Phase Shift (d)

1

State the phase shift of y = sin(x + 75°).

2

State the period and phase shift of y = cos(3x − 45°).

3

Complete: cos x = sin(x + ___).

Putting It All Together: Sketching y = a sin(k(x − d)) + c

1

For y = −2 sin(3(x + 20°)) + 4, state the amplitude, period, phase shift, axis and range.

2

List the key points of one cycle of y = −2 sin(3(x + 20°)) + 4.

3

Find the y-intercept of y = −2 sin(3(x + 20°)) + 4.

Writing the Equation of a Sinusoidal Function

1

Write a cosine equation for the graph in the second diagram of this review (maximum (30°, 2), minimum (120°, −4)).

2

Write a sine equation for the same graph.

3

Write an equation with amplitude 5, period 30, a minimum at x = 0 and axis y = 7.

Collecting Sinusoidal Data and Finding a Model

1

Data have a maximum of 16 at t = 1 h, a minimum of 4, and a period of 6 h. Write a model.

2

Give an example of data that are periodic but not sinusoidal.

3

Electricity in many European countries has a frequency of 50 Hz. What is the period in milliseconds?

Modelling Periodic Phenomena That Do Not Involve Angles

1

A Ferris wheel has radius 12 m, axle 14 m above the ground and period 48 s. A rider boards at the bottom. Write a model.

2

Use your model to find the height after 16 s.

3

State the domain and range of the model for a ride of 5 turns.

What If? Predicting How Changes Affect a Sinusoidal Model

1

A person walking in a circle in front of a motion sensor speeds up. Which parameter changes, and how?

2

A motion-sensor model is d = −2 cos(60t) + 4. Describe the circle and the walk.

3

The circle in the previous question is moved 1 m closer to the wall. Write the new model.

Solving Problems with Sinusoidal Models

1

For h = −12 cos(7.5t) + 14, when is the rider first 20 m high?

2

For the same model, how long during each turn is the rider above 20 m?

3

Find both angles from 0° to 360° with sin θ = 0.8 (to one decimal place).

Check your understanding

Quiz

1.

What is the amplitude of y = −6 cos 2x + 1?

2.

What is the period of y = sin 5x?

3.

What is the phase shift of y = sin(2x + 90°)?

4.

What is the range of y = 3 sin(x − 30°) − 2?

5.

Which function has a maximum of 7, a minimum of 1 and a period of 180°?

6.

What is sin 210°?

7.

A model of average monthly temperature has a period of 12 months. What is k?

8.

A Ferris wheel rider boards at the bottom at t = 0. Which form of model is simplest?

9.

On which intervals is y = sin x increasing for 0° ≤ x ≤ 360°?

10.

Tides at a harbour have a period of 12.4 h. High tide is at 3:00 a.m. When is the next high tide?

For parents and teachers

Use the review to find what to revisit:

  • Mixing up period, amplitude and axis, or using max − min as the amplitude → Lesson 1 (Periodic Functions).
  • Trouble predicting far-off values → Lesson 2 (Extrapolating Periodic Behaviour): subtract whole periods.
  • Wrong signs or exact values for angles beyond 90°, or calculator in radians → Lesson 3 (From the Sine Ratio to the Sine Function) and the unit Trigonometric Ratios and Laws.
  • Not knowing where sin x and cos x start, peak or cross zero → Lesson 4 (The Graphs of y = sin x and y = cos x).
  • Negative amplitudes, or adding c before multiplying by a → Lesson 5 (a and c).
  • Getting the period backwards (360° × k instead of 360° ÷ k) → Lesson 6 (k).
  • Shifting the wrong way, or not factoring before reading d → Lesson 7 (Phase Shift).
  • Errors in the mapping rule or incomplete properties → Lesson 8 (Putting It All Together).
  • Choosing d from a maximum while writing a sine equation → Lesson 9 (Writing the Equation).
  • Poor fits to data, or unrealistic domains → Lessons 10 and 11 (Collecting Data; Modelling Without Angles).
  • Not linking a change in the situation to the right parameter → Lesson 12 (What If?).
  • Missing the second solution, or answers without context → Lesson 13 (Solving Problems).

In Grade 12, Advanced Functions (MHF4U) revisits these functions with angles in radians, adds the tangent function and solves trigonometric equations more fully. Physics courses (SPH3U, SPH4U) use sinusoidal models for waves and sound.