Simple transformations and periodicity | Year 11

01 / LEARN

The idea, step by step

Separate the changes

For a nonconstant model $y = D + A sin (B (x - C))$ with $A \neq 0$ and $B \neq 0$ , the midline is D, amplitude is $∣ A ∣$ and period is $2 π / ∣ B ∣$ . The graph is shifted right by C. A negative A reflects the curve in its midline. Cosine follows the same feature rules; a nonconstant tangent model has period $π / ∣ B ∣$ and no amplitude.

T_{sin, cos} = 2 π / ∣ B ∣, T_{tan} = π / ∣ B ∣

Build and check

Find midline and amplitude from the maximum/minimum, then calculateB from the period. Choose sine or cosine to match the starting position. State the units of the horizontal variable. Equivalent-looking forms can represent the same curve, so verify a starting point and quarter-cycle points rather than judging appearance alone.

A common trap

Calling the multiplier inside sine its period.

Instead: That multiplier scales the input; divide 2π by its magnitude to obtain the period.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

For $y = 3 - 2 cos (4 x)$ , find amplitude, midline, period and range.

$∣ A ∣ = 2$ , D=3 and B=4.
$T = 2 π / 4 = π / 2$ .
The extremes are 3−2 and 3+2.

Amplitude 2; midline y=3; period π/2; range[1,5].

Example 2

Worked example 2

A sine curve has midline 1, amplitude 4, period π and crosses its midline upward at 0. Give a rule.

D=1, A=4.
$2 π / B = π$ gives B=2.
No shift is required for the upward crossing.

y = 1 + 4 sin (2 x)

Check your understanding

Find the period of sin(3 x).

Show answer

$2 π / 3$

Three cycles fit in the original 2π interval.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

State amplitude and range of $y = 5 + 3 sin x$ .
3 marks
Find the period of $y = cos (x / 2)$ .
3 marks
Describe the shift in $y = sin (x - π / 3)$ .
3 marks
For $y = tan (2 x)$ , find its period and closest asymptotes to 0.
3 marks
A cosine model has maximum 7, minimum−1, period 6 and starts at its maximum. Give a rule.
3 marks
Find the first maximum for x≥0 of $y = 2 sin (x - π / 4)$ .
3 marks

Read the worked solutions

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

State amplitude and range of $y = 5 + 3 sin x$ .
3 marks
Use midline 5 and vertical distance 3.
Amplitude 3; range[2,8].
Find the period of $y = cos (x / 2)$ .
3 marks
$2 π / (1 / 2) = 4 π$ .
$4 π$
Describe the shift in $y = sin (x - π / 3)$ .
3 marks
The argument becomes 0 at x=π/3.
π/3 to the right.
For $y = tan (2 x)$ , find its period and closest asymptotes to 0.
3 marks
The period isπ/2; solve $2 x = \pm π / 2$ for the boundaries.
Period π/2; asymptotes x=±π/4.
A cosine model has maximum 7, minimum−1, period 6 and starts at its maximum. Give a rule.
3 marks
Midline 3 and amplitude 4.
$B = 2 π / 6 = π / 3$ .
$y = 3 + 4 cos (π x / 3)$
Find the first maximum for x≥0 of $y = 2 sin (x - π / 4)$ .
3 marks
The argument isπ/2 at a maximum: $x - π / 4 = π / 2$ .
x=3π/4, value 2.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

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