Simple transformations and periodicity | Year 11

01 / LEARN

Understand the idea.

Separate the changes

In $y = D + A sin (B (x - C))$ , the midline is $y = D$ and the amplitude is $∣ A ∣$ . The inside factor changes the horizontal scale: one full sine cycle occurs when its argument changes by $2 π$ , giving period $\frac{2 π}{∣ B ∣}$ .

These feature rules require $A \neq 0$ and $B \neq 0$ . A negative $A$ reflects the graph across its midline, and $C$ shifts it right. Cosine has the same rules; tangent instead has period $\frac{π}{∣ B ∣}$ and no amplitude.

\begin{aligned} T_{sin, cos} & = \frac{2 π}{∣ B ∣} \\ T_{tan} & = \frac{π}{∣ B ∣} \end{aligned}

Build and check

Build a rule from its features: average the extreme values for the midline, halve their difference for the amplitude, and use the period to find the inside factor. Choose the sign and phase to match the starting height and direction of motion.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

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

Worked solution

1
Read the vertical features.
$D = 3, A = - 2, ∣ A ∣ = 2$
The negative sign reflects cosine but the amplitude is a positive distance from the midline.
2
Calculate the period.
$T = \frac{2 π}{∣ 4 ∣} = \frac{π}{2}$
The input $4 x$ completes a full turn four times as quickly as $x$ .
3
Find the range around the midline.
$3 - 2 \leq y \leq 3 + 2 ⟹ 1 \leq y \leq 5$
Cosine reaches both -1 and 1, so both endpoints occur.

Answer

Amplitude 2; midline y=3; period $\frac{π}{2}$ ; range[1,5].

EXAMPLE 02

Worked example 2

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

Worked solution

1
Set the vertical position and size.
$D = 1, A = 4$
A positive sine coefficient gives the required upward midline crossing.
2
Use the period to find the inside factor.
$\frac{2 π}{B} = π ⟹ B = 2$
Taking the positive factor gives a cycle of the required length.
3
Write and check the rule.
$y = 1 + 4 sin (2 x), y (0) = 1, y (\frac{π}{4}) = 5$
With no shift, the curve starts at the midline and reaches its maximum a quarter-period later.

Answer

$y = 1 + 4 sin (2 x)$

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the period of sin(3 x).

Show answer & reasoning

$\frac{2 π}{3}$

One cycle requires the argument $3 x$ to increase by $2 π$ . Thus $3 T = 2 π$ and $T = \frac{2 π}{3}$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

State amplitude and range of $y = 5 + 3 sin x$ .
3 marks
Find the period of $y = cos (\frac{x}{2})$ .
3 marks
Describe the shift in $y = sin (x - \frac{π}{3})$ .
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 $x \geq 0$ of $y = 2 sin (x - \frac{π}{4})$ .
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 $y = cos (\frac{x}{2})$ .
3 marks
$\frac{2 π}{\frac{1}{2}} = 4 π$ .
$4 π$
Describe the shift in $y = sin (x - \frac{π}{3})$ .
3 marks
The argument becomes 0 at $x = \frac{π}{3}$ .
$\frac{π}{3}$ to the right.
For $y = tan (2 x)$ , find its period and closest asymptotes to 0.
3 marks
The period is $\frac{π}{2}$ ; solve $2 x = \pm \frac{π}{2}$ for the boundaries.
Period $\frac{π}{2}$ ; asymptotes $x = \pm \frac{π}{4}$ .
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.
$B = \frac{2 π}{6} = \frac{π}{3}$ .
$y = 3 + 4 cos (\frac{π x}{3})$
Find the first maximum for $x \geq 0$ of $y = 2 sin (x - \frac{π}{4})$ .
3 marks
The argument is $\frac{π}{2}$ at a maximum: $x - \frac{π}{4} = \frac{π}{2}$ .
$x = \frac{3 π}{4}$ , 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 ↗

In this lesson

Understand the idea.

Separate the changes

Build and check

See the method in action.

Worked example 1

Read the vertical features.

Calculate the period.

Find the range around the midline.

Worked example 2

Set the vertical position and size.

Use the period to find the inside factor.

Write and check the rule.

Try it yourself.

Check your understanding

Keep practising