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D. Trigonometric Functions · Sinusoidal Functions

Transformations III: Horizontal Translations and Phase Shift (d)

D2.5 D2.6 D2.7About 70 minutes
Learning goals
  • I can explain how the parameter d in y = sin(x − d) and y = cos(x − d) changes the graph.
  • I can find the phase shift of y = a sin(k(x − d)) + c, factoring k out when needed.
  • I can sketch a graph with a phase shift using key points.
  • I can explain why a sinusoidal graph can be written with more than one phase shift.

Words to know

horizontal translation
Sliding a graph left or right without changing its shape.
phase shift
The horizontal translation of a sinusoidal function: d in y = a sin(k(x − d)) + c. Right if d > 0, left if d < 0.
factored form
Writing the inside of the function as k(x − d), for example 2x − 60° = 2(x − 30°), so the phase shift can be read directly.

Same wave, later start

Two friends board the same Ferris wheel, one car apart. Their height graphs have exactly the same shape and the same period. The only difference is when things happen: the second friend reaches the top a few seconds later. On a graph, that is a horizontal translation. For sinusoidal functions it has a special name: the phase shift.

What d does

In y = sin(x − d), the graph of y = sin x is translated d units to the right (or |d| units to the left if d is negative).

Why does a minus sign mean right? The maximum of sine happens when its input is 90°. For y = sin(x − 60°) the input is x − 60°, so the maximum happens when x − 60° = 90°, that is, at x = 150°, which is 60° to the right of 90°. Everything happens 60° later.

Mapping rule: (x, y) → (x + d, y)

  • y = sin(x − 60°): d = 60, shift 60° right.
  • y = cos(x + 45°) = cos(x − (−45°)): d = −45, shift 45° left.

This matches Grade 10, where y = (x − h)² was y = x² moved h units right.

0°60°120°180°240°300°360°420°−101260° right60° right(150°, 1)(60°, 0)(330°, −1)
y = sin(x − 60°) (solid) is y = sin x (dashed) translated 60° to the right. Every key point is 60° later.
Worked example

Example 1: Sketching y = sin(x − 60°)

List the key points of one cycle of y = sin(x − 60°).

  1. d = 60: add 60° to each x-value of the sine key points.
  2. (0°, 0) → (60°, 0); (90°, 1) → (150°, 1); (180°, 0) → (240°, 0); (270°, −1) → (330°, −1); (360°, 0) → (420°, 0).
  3. The period (360°), amplitude (1) and axis (y = 0) do not change.

(60°, 0), (150°, 1), (240°, 0), (330°, −1), (420°, 0)

Worked example

Example 2: A shift to the left

Sketch one cycle of y = cos(x + 45°).

  1. Rewrite: x + 45° = x − (−45°), so d = −45: shift 45° left.
  2. Subtract 45° from each x-value of the cosine key points (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1):
  3. (−45°, 1), (45°, 0), (135°, −1), (225°, 0), (315°, 1).

(−45°, 1), (45°, 0), (135°, −1), (225°, 0), (315°, 1)

Watch out when k is not 1

In y = sin(2x − 60°) it is tempting to say "shift 60° right". That is wrong. The phase shift is d in k(x − d), so first factor out k:

2x − 60° = 2(x − 30°)

So y = sin(2x − 60°) = sin(2(x − 30°)): period 180°, phase shift 30° right.

The full mapping rule for y = a sin(k(x − d)) + c is

(x, y) → (x/k + d, ay + c)

Divide by k first, then add d.

Worked example

Example 3: Factoring to find the phase shift

Find the period and phase shift of y = sin(2x − 60°) and list the key points of one cycle.

  1. Factor: y = sin(2(x − 30°)). k = 2, d = 30.
  2. Period 360° ÷ 2 = 180°. Phase shift 30° right.
  3. Apply (x, y) → (x/2 + 30°, y): (0°, 0) → (30°, 0); (90°, 1) → (75°, 1); (180°, 0) → (120°, 0); (270°, −1) → (165°, −1); (360°, 0) → (210°, 0).
  4. Check one point in the original equation: x = 75° gives sin(150° − 60°) = sin 90° = 1 ✓.

Period 180°, phase shift 30° right; key points (30°, 0), (75°, 1), (120°, 0), (165°, −1), (210°, 0).

0°30°60°90°120°150°180°210°240°−1012(75°, 1)(165°, −1)(30°, 0)correct: y = sin(2(x − 30°))wrong: shifted 60°
The correct graph of y = sin(2x − 60°) = sin(2(x − 30°)) (solid) starts its cycle at 30°. The dashed graph shows the common mistake of shifting 60°.

More than one right answer

Shifting a sine wave by one full period lands it on itself. So d and d ± 360° (or, in general, d plus or minus any multiple of the period) give the same graph. For example,

sin(x − 30°) = sin(x + 330°)

And since cosine is sine shifted 90° left, any sinusoidal graph can be written as either a sine or a cosine:

cos x = sin(x + 90°) and sin x = cos(x − 90°)

When you write equations later in this unit, there will usually be several correct answers. Pick the one with the simplest phase shift.

Worked example

Example 4: Same graph, different equations

Show that y = sin(x − 90°) and y = −cos x have the same graph.

  1. Key points of y = sin(x − 90°): add 90° to each sine key point: (90°, 0), (180°, 1), (270°, 0), (360°, −1), (450°, 0). Since the graph repeats every 360°, the point (360°, −1) also gives (0°, −1).
  2. Key points of y = −cos x: (0°, −1), (90°, 0), (180°, 1), (270°, 0), (360°, −1).
  3. Both graphs pass through (0°, −1), (90°, 0), (180°, 1), (270°, 0) and (360°, −1), and both have period 360°, so they are the same.

sin(x − 90°) = −cos x.

Just for fun

Phase shift is the math name for being fashionably late. The graph of y = sin(x − 30°) does everything y = sin x does, just 30° later, and it never apologizes.

Hands-on activity

Do "the wave" at home

You need: 3 or 4 people, chairs in a row and a timer (or someone counting seconds).

  1. Everyone sits. The first person stands up and sits down at a steady pace: up for 2 seconds, down for 2 seconds, over and over (period 4 s).
  2. The second person does exactly the same thing, but starts 1 second later. The third starts 1 second after the second, and so on.
  3. Watch from the side: a wave travels along the row!
  4. Sketch the graph of head height against time for person 1 and for person 3 on the same axes.
  5. Answer: What is the period of each graph? How are the graphs of person 1 and person 3 related? Which parameter, a, k, d or c, describes the difference?

What to look for: every graph has the same shape, period (4 s) and amplitude. Person 3's graph is person 1's graph translated 2 seconds to the right: a phase shift, d = 2. (Standing and sitting give a graph with flat tops, so it is periodic but only roughly sinusoidal: a nice reminder from the first lesson.)

Your turn

1

State the phase shift of y = sin(x − 40°).

2

State the phase shift of y = cos(x + 30°).

3

List the key points of one cycle of y = sin(x + 90°). Which parent function has the same graph?

4

Find the period and phase shift of y = sin(3x − 90°).

5

Find the period and phase shift of y = cos(2x + 180°).

6

Write the equation of y = sin x translated 45° right and 2 units up.

7

Where does the point (0°, 1) on y = cos x go under y = cos(x − 120°)?

More practice: warm-up

1

Describe the phase shift of y = sin(x − 25°).

2

Describe the phase shift of y = cos(x + 10°).

3

Write the equation of y = sin x translated 70° right.

4

Does the phase shift change the period?

5

Does the phase shift change the range?

More practice: on your own

1

List the key points of one cycle of y = cos(x − 60°).

2

Find the period and phase shift of y = sin(4x + 120°).

3

For y = 2 sin(x − 30°) + 1, find the maximum value and the first x ≥ 0° where it happens.

4

Give two different values of d, one positive and one negative, for which y = sin(x − d) has the same graph as y = sin(x − 100°).

5

Explain why sin(x − 360°) = sin x for every x.

Challenge

1

Find the smallest positive d for which y = cos(x − d) has the same graph as y = sin x.

2

Write y = −sin x as a sine function with a phase shift (no negative a).

3

Find the period and phase shift of y = sin(0.5x − 45°), and list the key points of one cycle.

Check your understanding

Quiz

1.

What is the phase shift of y = sin(x + 50°)?

2.

What is the phase shift of y = cos(3x − 120°)?

3.

Under y = sin(x − 30°), where does the point (90°, 1) of y = sin x go?

4.

Which equation translates y = cos x by 60° to the right?

5.

Which function has the same graph as y = cos x?

6.

Which of these does the phase shift d change?

For parents and teachers

Big idea: This lesson completes the four parameters for D2.5 to D2.7 with d, the phase shift. The two big ideas are "minus means right" and factor out k before reading d.

How to help: Ask your teen to explain why y = sin(x − 60°) reaches its maximum at 150° ("the inside has to be 90°"). For any equation with a number in front of x, ask, "Did you factor?"

Common difficulties:

  • Shifting the wrong way: (x − 60°) means right, (x + 60°) means left.
  • Not factoring: sin(2x − 60°) shifts 30°, not 60°.
  • Applying the mapping rule in the wrong order. Divide by k first, then add d.
  • Being surprised by multiple correct answers. Different phase shifts that differ by a full period (or sine versus cosine) can describe the same graph.