B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Amplitude, Period and Phase Shift in Radians

B2.4About 70 minutes

Learning goals

I can find the amplitude, period, phase shift and vertical translation of y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c, with radian measures.
I can use period = 2π/|k| and find k from a given period.
I can factor an equation like y = sin(2x − π/3) to find the correct phase shift.
I can find the maximum, minimum and range from the equation.

Words to know

amplitude
|a|: half the distance from the maximum to the minimum.

period
2π/|k| when x is in radians: the horizontal length of one cycle.

phase shift
The horizontal translation d of a sinusoidal function: right if d > 0, left if d < 0.

vertical translation
The value c; the axis of the curve is the line y = c.

sinusoidal function
A function whose graph has the shape of a sine wave: y = a sin(k(x − d)) + c or y = a cos(k(x − d)) + c.

The four parameters, in radians

Remember from Grade 11: in Sinusoidal Functions you met the transformed functions

y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c

with x in degrees, and the period was 360°/|k|. Everything works the same way in radians except the period:

a: vertical stretch by a factor of |a|. Amplitude = |a|. If a < 0, the graph is also reflected in the x-axis.
k: horizontal compression or stretch by a factor of 1/|k|. Period = 2π/|k|. If k < 0, the graph is also reflected in the y-axis.
d: horizontal translation, the phase shift. Right if d > 0, left if d < 0.
c: vertical translation. The axis is y = c.

The maximum is c + |a| and the minimum is c − |a|, so the range is {y ∈ ℝ | c − |a| ≤ y ≤ c + |a|}.

Replacing x with 2x squeezes the graph horizontally by a factor of 1/2. Two full cycles of y = sin 2x fit where y = sin x makes one, so its period is 2π ÷ 2 = π.

Why the period is 2π/|k|

y = sin x completes one cycle as x goes from 0 to 2π. For y = sin(kx) with k > 0, the input kx must go from 0 to 2π, so x goes from 0 to 2π/k. Another way to think about it: k tells you how many cycles fit in a 2π interval.

To find k from a period: k = 2π ÷ period. For example, a period of 12 (months) gives k = 2π/12 = π/6.

Subtracting π/4 inside the bracket moves every point π/4 to the right. The maximum moves from π/2 to 3π/4.

Factor out k before reading d

The phase shift is d only when the bracket is written as k(x − d). In y = sin(2x − π/3), the phase shift is not π/3. Factor out the 2 first: 2x − π/3 = 2(x − π/6). The phase shift is π/6 to the right.

Worked example

Example 1: Read all four parameters

For y = 4 sin(3(x − π/6)) − 2, state the amplitude, period, phase shift, vertical translation, maximum, minimum and range.

a = 4, k = 3, d = π/6, c = −2.
Amplitude |a| = 4. Period 2π/3. Phase shift π/6 to the right. Vertical translation 2 down; axis y = −2.
Maximum −2 + 4 = 2; minimum −2 − 4 = −6.

Amplitude 4, period 2π/3, phase shift π/6 right, axis y = −2, max 2, min −6, range {y ∈ ℝ | −6 ≤ y ≤ 2}.

Worked example

Example 2: Factor first

Find the period and phase shift of y = −2 cos(x/2 + π/4) + 1, and describe the reflection.

Factor out k = 1/2: x/2 + π/4 = (1/2)(x + π/2). (Check: (1/2)(π/2) = π/4 ✓.)
So k = 1/2 and d = −π/2.
Period = 2π ÷ (1/2) = 4π. Phase shift π/2 to the left.
a = −2: amplitude 2, reflected in the x-axis. Axis y = 1.

Period 4π, phase shift π/2 left, amplitude 2 with a reflection in the x-axis, axis y = 1.

Worked example

Example 3: A period that is not a multiple of π

A model for a household's hourly electricity use is y = 5 sin(πx/12) + 10, where x is in hours. State the period and amplitude.

k = π/12.
Period = 2π ÷ (π/12) = 2π × 12/π = 24.
Amplitude 5; axis y = 10; max 15; min 5.

Period 24 hours (one day), amplitude 5. When x is time, the period is a number of hours, not an angle; this is why radians are so convenient in models.

Worked example

Example 4: Find k from the period

A sinusoidal function has period π/2. Find k (k > 0). Then write a cosine function with this period, amplitude 3 and axis y = 4.

k = 2π ÷ period = 2π ÷ (π/2) = 4.
a = 3, c = 4, and no phase shift is needed.

k = 4; for example y = 3 cos(4x) + 4.

y = 2 sin(2(x − π/4)) + 1 with its parameters marked: the axis y = 1, the amplitude 2, the period π, and a cycle that starts at x = π/4 instead of at 0.

Just for fun

A sine wave with a phase shift is like a friend who is always exactly π/4 late. Annoying, but at least you can predict it.

Hands-on activity

Slider investigation

You need: a computer, tablet or phone with a free online graphing calculator, and paper to record results.

Type y = a sin(k(x − d)) + c. Most graphing tools will offer to make sliders for a, k, d and c. Set the x-axis to show from about −1 to 13 (or use radian/π gridlines if your tool has them).
Start with a = 1, k = 1, d = 0, c = 0. Move only a. Record what changes and what stays fixed.
Reset, then move only k. Find the k-values that give exactly 2, 3 and 1/2 cycles between 0 and 2π. Record each period.
Reset, then move only d. When d = π/2 (about 1.57), what familiar graph do you get?
Reset, then move only c.
Now type y = sin(2x − 1) and y = sin(2(x − 0.5)). Do the graphs match? Explain.

What to look for: a changes the height (and flips the graph when negative); k = 2, 3, 0.5 give periods π, 2π/3 and 4π; d slides the graph sideways, and d = π/2 turns y = sin x into y = sin(x − π/2), which is the cosine graph flipped upside down (y = −cos x); c moves the axis up or down. The two equations in step 6 give the same graph, because 2x − 1 = 2(x − 0.5).

Your turn

State the amplitude and period of y = 3 sin 2x.

State the phase shift and the equation of the axis of y = cos(x − π/3) + 4.

Describe the graph of y = −5 sin(x/3) compared with y = sin x.

For y = 2 cos(4(x + π/8)) − 1, state the amplitude, period, phase shift and range.

Find the period and phase shift of y = sin(2x − π/3).

Find k > 0 so that y = sin(kx) has period 3π.

For y = 6 cos(πx/4) + 2, find the period, maximum and minimum.

More practice: warm-up

Period of y = sin 4x?

Amplitude of y = −7 cos x?

Phase shift of y = sin(x + π/2)?

Axis of y = cos x − 3?

Period of y = cos(x/2)?

More practice: on your own

For y = 0.5 sin(3x + π) + 2, find the amplitude, period, phase shift and range.

A sinusoidal function has period π/3. Find k (k > 0).

Find the maximum, minimum and range of y = −3 sin(2(x − π/4)) + 5.

A student says the phase shift of y = cos(4x − π) is π to the right. Correct the error.

A model for the average monthly temperature (°C) in a Canadian city is T = 13 sin((π/6)(t − 4)) + 8, where t is the month number (t = 1 for January). State the period, amplitude, phase shift, maximum and minimum, and say what each means.

Challenge

Find the period and phase shift of y = sin(−2x + π), and describe all the transformations.

Explain why the phase shift of a sinusoidal function is not unique. Give two other phase shifts that describe y = sin(x − π/4).

Let f(x) = 3 sin 2x and g(x) = 3 sin(2x − π/2). By how much is g shifted relative to f, as a fraction of the period? Show that g(x) = −3 cos 2x at x = 0 and x = π/4.

Check your understanding

Quiz

What is the period of y = 3 sin(4x) − 1?

What is the phase shift of y = cos(2x − π/2)?

What is the amplitude of y = −6 sin(x/2) + 1?

What is the range of y = 2 cos(x − π) + 3?

Which function has period 6?

In y = a sin(k(x − d)) + c, which parameter changes the period?

For parents and teachers

Big idea (B2.4): in y = a sin(k(x − d)) + c, the amplitude is |a|, the period is 2π/|k|, the phase shift is d and the axis is y = c. The only change from Grade 11 is 2π instead of 360° in the period.

How to help: Give your teen an equation like y = 2 sin(3x − π) + 1 and ask them to say the four properties out loud. Listen for "factor out the 3 first". A free online graphing calculator lets them check by graphing.

Common difficulties:

Reading the phase shift without factoring out k.
Writing the period as 2πk instead of 2π/k.
Giving a negative amplitude.
Getting the direction of the phase shift backwards: x − d moves right, x + d moves left.