Transformed sinusoidal models including phase shift | Year 11

01 / LEARN

The idea, step by step

Identify vertical and horizontal changes

For $y = A sin (B (x - h)) + D$ , the amplitude is $∣ A ∣$ and the midline is $y = D$ . Its maximum and minimum are $D + ∣ A ∣$ and $D - ∣ A ∣$ . Negative $A$ reflects the familiar shape about the midline.

The period is $2 π / ∣ B ∣$ and $h$ is the horizontal shift. Factor the argument first: $2 x - π = 2 (x - π / 2)$ . This is a right shift of $π / 2$ , not $π$ . The period formula assumes $B \neq 0$ ; if $B = 0$ , the rule is constant and does not have a unique fundamental period.

T = \frac{2 π}{∣ B ∣}

Sketch and construct

Mark the midline and use quarter-period steps to place crossings and extrema. Sine begins on its midline at zero argument; a positive cosine begins at a maximum. The graph below can also be written $y = 1 - 2 cos x$ .

To construct a rule, take half the difference of maximum and minimum for amplitude and their average for the midline. Use the gap between corresponding points for period. Equivalent sine and cosine rules are acceptable if they match all the given features.

A common trap

Reading the constant inside an unfactored argument as the phase shift.

Instead: Rewrite the argument as $B (x - h)$ before identifying $h$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Describe $y = 3 cos (2 (x - π / 4)) - 1$ .

Amplitude is 3; midline is $y = - 1$ , giving range $[- 4, 2]$ .
Period is $2 π / 2 = π$ .
The horizontal shift is $π / 4$ right; a maximum occurs at $x = π / 4$ .

Amplitude 3; period

π

; shift right

π / 4

; range

[- 4, 2]

Example 2

A sinusoid has maximum 7, minimum 1 and consecutive maxima at $x = 2$ and $x = 10$ . Give a cosine rule.

Midline 4 and amplitude 3 follow from the extreme values.
Period 8 gives $B = π / 4$ .
Shift a positive cosine maximum to $x = 2$ .

y = 4 + 3 cos (\frac{π}{4} (x - 2))

Check your understanding

What is the period of $- 2 sin (3 x) + 5$ ?

Show answer

$2 π / 3$ .

Only the coefficient of $x$ changes the period.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

State the amplitude and range of $4 sin x - 2$ .
2 marks
Find the period of $2 cos (x / 3)$ .
2 marks
Find the horizontal shift of $sin (4 x - π)$ .
2 marks
A cosine graph has range $[- 5, 3]$ , period $2 π$ and a maximum at 0. Give a rule.
2 marks
Find the first two maxima of $2 cos (π x / 3)$ for $x \geq 0$ .
2 marks
Explain the difference between $sin x + 4$ and $4 sin x$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

State the amplitude and range of $4 sin x - 2$ .
2 marks
The midline is $- 2$ ; add and subtract 4.
Amplitude 4; range $[- 6, 2]$ .
Find the period of $2 cos (x / 3)$ .
2 marks
$B = 1 / 3$ , so $T = 2 π / (1 / 3)$ .
$6 π$ .
Find the horizontal shift of $sin (4 x - π)$ .
2 marks
Factor the argument as $4 (x - π / 4)$ .
Right $π / 4$ .
A cosine graph has range $[- 5, 3]$ , period $2 π$ and a maximum at 0. Give a rule.
2 marks
The mean of the endpoints is $- 1$ , half their difference is 4, and $B = 1$ .
$y = - 1 + 4 cos x$ .
Find the first two maxima of $2 cos (π x / 3)$ for $x \geq 0$ .
2 marks
Period 6; positive cosine starts at a maximum.
$x = 0, 6$ .
Explain the difference between $sin x + 4$ and $4 sin x$ .
2 marks
Their ranges are $[3, 5]$ and $[- 4, 4]$ , respectively.
The first changes the midline; the second changes amplitude.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

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