Grade by Grade
D. Trigonometric Functions · Sinusoidal Functions

Transformations II: Horizontal Stretches and the Period (k)

D2.5 D2.6 D2.7About 70 minutes
Learning goals
  • I can explain how the parameter k in y = sin(kx) and y = cos(kx) changes the graph.
  • I can find the period from an equation using period = 360° ÷ |k|.
  • I can find k when I know the period.
  • I can sketch graphs that combine a, k and c.

Words to know

horizontal compression
Squeezing a graph toward the y-axis. In y = sin(kx) with k > 1, every x-value is divided by k.
horizontal stretch
Pulling a graph away from the y-axis. In y = sin(kx) with 0 < k < 1, every x-value is divided by k, which makes it larger.
period formula
For y = a sin(k(x − d)) + c or y = a cos(k(x − d)) + c, period = 360° ÷ |k|.
frequency
The number of cycles in one unit (for example, cycles per second). Frequency = 1 ÷ period.

Faster waves

Play a low note and a high note on a guitar. The high-note string vibrates faster: it completes more cycles every second, so each cycle is shorter. On a graph, faster vibration means the waves are squeezed closer together horizontally.

The parameter that controls this is k in y = sin(kx) and y = cos(kx).

What k does

Compare y = sin x and y = sin 2x. To reach the maximum, sin needs an input of 90°. In y = sin 2x the input is 2x, so the maximum comes when 2x = 90°, that is, at x = 45°. Everything happens twice as soon.

In general, the graph of y = sin(kx) is the graph of y = sin x with every x-value divided by k:

mapping rule: (x, y) → (x/k, y)

  • If k > 1, the graph is compressed horizontally by a factor of 1/k (waves closer together).
  • If 0 < k < 1, the graph is stretched horizontally by a factor of 1/k (waves farther apart).

One cycle of sin x takes 360°, so one cycle of sin(kx) takes 360° ÷ k:

period = 360° ÷ |k| and so k = 360° ÷ period.

The amplitude and the axis do not change; only the x-values do. (If k is negative, there is also a reflection in the y-axis; in this course, you can always choose a positive k.)

0°45°90°135°180°225°270°315°360°−1012period of y = sin 2x: 180°(45°, 1)(135°, −1)y = sin x (dashed)y = sin 2x
y = sin 2x (solid) fits a full cycle into 180°, so two cycles fit where y = sin x (dashed) has one. Its key points are 45° apart.
Worked example

Example 1: Sketching y = sin 2x

State the period of y = sin 2x and list the key points of one cycle.

  1. k = 2, so the period is 360° ÷ 2 = 180°.
  2. Key points are a quarter period apart: 180° ÷ 4 = 45°.
  3. Apply (x, y) → (x/2, y) to (0°, 0), (90°, 1), (180°, 0), (270°, −1), (360°, 0):
  4. (0°, 0), (45°, 1), (90°, 0), (135°, −1), (180°, 0). The pattern zero, max, zero, min, zero is unchanged.

Period 180°; key points (0°, 0), (45°, 1), (90°, 0), (135°, −1), (180°, 0).

Worked example

Example 2: A horizontal stretch

Sketch one cycle of y = cos(0.5x).

  1. k = 0.5, so the period is 360° ÷ 0.5 = 720°: the wave is stretched horizontally by a factor of 2.
  2. Quarter period: 720° ÷ 4 = 180°.
  3. Apply (x, y) → (x ÷ 0.5, y) = (2x, y) to the cosine key points (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1):
  4. (0°, 1), (180°, 0), (360°, −1), (540°, 0), (720°, 1).

Period 720°; key points (0°, 1), (180°, 0), (360°, −1), (540°, 0), (720°, 1).

0°90°180°270°360°450°540°630°720°−1012period of y = cos(0.5x): 720°(360°, −1)(180°, 0)(540°, 0)y = cos x (dashed)
y = cos(0.5x) (solid) is y = cos x (dashed) stretched horizontally by a factor of 2. One cycle now takes 720°.
Worked example

Example 3: Finding k from the period

Find k (k > 0) for a sinusoidal function with period (a) 120° (b) 24 (hours) (c) 1080°.

  1. Use k = 360° ÷ period.
  2. (a) k = 360 ÷ 120 = 3.
  3. (b) k = 360 ÷ 24 = 15. (When x is a time, k is measured in degrees per hour. You will use this in the lesson Modelling Periodic Phenomena That Do Not Involve Angles.)
  4. (c) k = 360 ÷ 1080 = 1/3.

k = 3, 15 and 1/3.

Worked example

Example 4: Combining a, k and c

Sketch one cycle of y = 2 sin 3x + 1 and state its properties.

  1. a = 2: amplitude 2. k = 3: period 360° ÷ 3 = 120°. c = 1: axis y = 1.
  2. Mapping rule: (x, y) → (x/3, 2y + 1).
  3. (0°, 0) → (0°, 1); (90°, 1) → (30°, 3); (180°, 0) → (60°, 1); (270°, −1) → (90°, −1); (360°, 0) → (120°, 1).
  4. Range: {y ∈ ℝ | −1 ≤ y ≤ 3}. Domain: {x ∈ ℝ}.

Key points (0°, 1), (30°, 3), (60°, 1), (90°, −1), (120°, 1); amplitude 2, period 120°, axis y = 1.

Divide, don't multiply

In y = sin(kx), the x-values are divided by k. A bigger k gives a shorter period. It feels backwards at first, so always check with the period formula: y = sin 4x has period 360° ÷ 4 = 90°, much shorter than 360°.

Just for fun

The A above middle C on a piano is usually tuned to vibrate 440 times per second, and the A one octave higher vibrates 880 times per second. Your ears hear "k doubled, period halved" as the same note, only higher.

Hands-on activity

The paper-strip wave machine

You need: a long strip of paper (tape 4 or 5 sheets end to end, or use a roll of receipt paper), a marker, a ruler, a timer and a partner.

  1. Lay the strip on a table. Draw a straight line along the middle of it (the axis).
  2. Hold the marker on the line. Move it side to side across the line, about 5 cm each way, at a steady rhythm (say, one full back-and-forth every second). At the same time, your partner pulls the strip slowly and steadily underneath it. A wave appears!
  3. Make a second wave. This time move the marker back and forth twice as fast, while your partner pulls at the same speed.
  4. Make a third wave with your normal rhythm, while your partner pulls twice as fast.
  5. Measure the length of one cycle on each wave.

What to look for: doubling your rhythm should make each cycle about half as long (like k = 2). Pulling the paper faster makes each cycle longer (like 0 < k < 1). The height of the waves (the amplitude) stays about the same, because your hand moved the same distance each time.

Your turn

1

State the period of y = sin 4x.

2

State the period of y = cos(x/3).

3

List the key points of one cycle of y = sin 3x.

4

Find k (k > 0) if the period is 45°.

5

For y = 5 cos 2x + 1, state the amplitude, period, axis and range.

6

Describe how y = sin(0.25x) is related to y = sin x, and give its period.

7

Where does the point (90°, 1) on y = sin x go under y = sin 6x?

More practice: warm-up

1

State the period of y = sin 2x.

2

State the period of y = cos 5x.

3

State the period of y = sin(0.5x).

4

If k = 10, what is the period?

5

Does changing k change the amplitude?

More practice: on your own

1

For y = 3 sin 2x − 1, list the key points of one cycle and state the range.

2

For y = −2 cos 4x, state the period and list the key points of one cycle.

3

A function y = 3 sin(kt) has a period of 15 s. Find k and the first time after t = 0 when y is a maximum.

4

How many complete cycles of y = sin 3x are there for 0° ≤ x ≤ 360°?

5

Find the zeros of y = sin 2x for 0° ≤ x ≤ 360°.

Challenge

1

For what positive k does y = cos(kx) have exactly 5 complete cycles between 0° and 360°?

2

The graph of y = sin(kx), k > 0, has its first maximum to the right of the origin at (20°, 1). Find k and the period.

3

Show that y = sin(−2x) and y = −sin 2x have the same graph. What transformation of y = sin 2x is this?

Check your understanding

Quiz

1.

What is the period of y = sin 3x?

2.

What is the period of y = cos(0.2x)?

3.

A sinusoidal function has a period of 90°. What is k (k > 0)?

4.

How does y = sin 2x compare with y = sin x?

5.

Which function has a period of 720°?

6.

Under y = sin 4x, where does the point (180°, 0) of y = sin x go?

For parents and teachers

Big idea: This lesson continues D2.5 to D2.7 with the parameter k. The key result is period = 360° ÷ |k|. A larger k gives a shorter period.

How to help: Quiz your teen with quick pairs: "k = 2?" (180°), "period 30°?" (k = 12). Connect to music or to a ceiling fan on high and low speed: faster spinning means a shorter period.

Common difficulties:

  • Multiplying x by k instead of dividing. y = sin 2x reaches its maximum at 45°, not at 180°.
  • Thinking k changes the amplitude.
  • Forgetting to recompute the spacing of the key points: they are a quarter of the new period apart.
  • Mixing up k and the period in word problems. Write "period = …" first, then compute k = 360 ÷ period.