Transformed sinusoidal models including phase shift | Year 11

01 / LEARN

Understand the idea.

Identify vertical and horizontal changes

For a nonconstant sinusoid $y = A sin (B (x - h)) + D$ , with $A \neq 0$ and $B \neq 0$ , amplitude is $∣ A ∣$ and the midline is $y = D$ . Its range is $[D - ∣ A ∣, D + ∣ A ∣]$ . A negative $A$ reflects the sinusoid in its midline.

The period of this nonconstant sinusoid is $\frac{2 π}{∣ B ∣}$ . Factor before reading the shift: $2 x - π = 2 (x - \frac{π}{2})$ shifts right by $\frac{π}{2}$ . If $A = 0$ or $B = 0$ , the rule is instead constant at $y = D$ , with range ${D}$ and no fundamental period.

T = \frac{2 π}{∣ B ∣}

Sketch and construct

Locate a convenient starting phase, then mark points a quarter-period apart. Positive sine starts at a rising midline crossing; positive cosine starts at a maximum. The illustrated graph also has the equivalent form $y = 1 - 2 cos x$ .

To construct a model, average the maximum and minimum for the midline and halve their difference for amplitude. Measure the period between corresponding points, such as consecutive maxima. Then choose a phase that places a known crossing or extremum correctly.

Graph of y=2sin(x minus pi/2)+1, equivalently y=1 minus 2cos(x), for x from zero to two pi radians. Quarter-period ticks use stacked fractions; the dashed gold line is y=1.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Example 1

Describe $y = 3 cos (2 (x - \frac{π}{4})) - 1$ .

Worked solution

1
Read the vertical features.
$A = 3, D = - 1, - 1 - 3 \leq y \leq - 1 + 3$
Cosine ranges from $- 1$ to $1$ , giving amplitude $3$ , midline $- 1$ and range $[- 4, 2]$ .
2
Calculate the period.
$T = \frac{2 π}{2} = π$
Multiplying the input by two halves the usual cosine period.
3
Read the phase from the factored argument.
$h = \frac{π}{4}, y (\frac{π}{4}) = 2$
The shift is right by $\frac{π}{4}$ ; positive cosine has its maximum at zero argument.

Answer

Amplitude 3; period $π$ ; shift right $\frac{π}{4}$ ; range $[- 4, 2]$ .

EXAMPLE 02

Example 2

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

Worked solution

1
Calculate midline and amplitude.
$D = \frac{7 + 1}{2} = 4, A = \frac{7 - 1}{2} = 3$
The midpoint of the extremes is the midline, and their half-difference is the amplitude.
2
Calculate the frequency coefficient.
$T = 10 - 2 = 8, B = \frac{2 π}{8} = \frac{π}{4}$
Consecutive maxima are one full period apart.
3
Place a maximum at the given input.
$y = 4 + 3 cos (\frac{π}{4} (x - 2))$
At $x = 2$ and $x = 10$ the arguments are $0$ and $2 π$ , so both are maxima of height $7$ .

Answer

$y = 4 + 3 cos (\frac{π}{4} (x - 2))$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$\frac{2 π}{3}$ .

The input coefficient is $3$ , so $T = \frac{2 π}{3}$ . The factor $- 2$ changes amplitude and orientation, and $5$ shifts the midline; neither changes the period.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

State the amplitude and range of $4 sin x - 2$ .
2 marks
Find the period of $2 cos (\frac{x}{3})$ .
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 $2 cos (\frac{π x}{3})$ 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 $2 cos (\frac{x}{3})$ .
2 marks
$B = \frac{1}{3}$ , so $T = \frac{2 π}{\frac{1}{3}}$ .
$6 π$ .
Find the horizontal shift of $sin (4 x - π)$ .
2 marks
Factor the argument as $4 (x - \frac{π}{4})$ .
Right $\frac{π}{4}$ .
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 $2 cos (\frac{π x}{3})$ 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.

Official curriculum source ↗

In this lesson

Understand the idea.

Identify vertical and horizontal changes

Sketch and construct

See the method in action.

Example 1

Read the vertical features.

Calculate the period.

Read the phase from the factored argument.

Example 2

Calculate midline and amplitude.

Calculate the frequency coefficient.

Place a maximum at the given input.

Try it yourself.

Check your understanding

Keep practising