The idea, step by step
Start with one complete cycle
The sine curve passes through (0°,0), rises to (90°,1), returns to zero at 180°, reaches −1 at 270° and returns to zero at 360°. Cosine starts at its maximum: (0°,1). Both repeat after 360°.
Use key points and a smooth curve; do not connect them with sharp straight segments. A sketch needs labelled axes, scale, intercepts and turning points.
Read the parameters
For y = A sin(Bx) + D or A cos(Bx) + D in degrees, with A and B nonzero, amplitude is |A|, period is 360°/|B| and the midline is y = D. The range goes from D − |A| to D + |A|. If A or B is zero, the rule is constant and has no fundamental period.
A negative A reflects the graph in its midline. Cosine is even: cos(−x) = cos x. Sine is odd: sin(−x) = −sin x. Use the graph or unit circle to justify these symmetries.
A common trap
Multiplying the basic period by B.
Instead: A larger |B| fits more cycles into the same x interval, so divide 360° by |B|.
See the thinking
Read a transformed rule
State amplitude, period, midline and range for y = 2 sin x − 1.
- Amplitude is |2| = 2.
- B = 1, so the period is 360°.
- D = −1, so the midline is y = −1 and the range is −3 to 1.
A compressed cosine graph
For y = 3 cos(2x), state the period and the first maximum and minimum at or after x = 0°.
- The amplitude is 3.
- The period is 360°/2 = 180°.
- Cosine is 1 when 2x = 0° and −1 when 2x = 180°.
Check your understanding
Find the amplitude and period of y = 4 sin(2x) + 1.
Show answer
Amplitude 4; period 180°.
Amplitude is |4|; period is 360°/2.
Your turn
6 questions · 17 marks · Work at your own pace.
Read the questions online
- State the amplitude, period and range of y = 4 sin(2x) + 1.3 marks
- Give y = cos x at x = 0°, 90°, 180°, 270° and 360°.2 marks
- Sketch y = sin x for 0° ≤ x ≤ 360° on the supplied grid. Label the intercepts and turning points.4 marks
- Describe how y = −2 cos x differs from y = 2 cos x.2 marks
- Explain why sin(360° − x) = −sin x using the unit circle.3 marks
- A sine graph has maximum 3, minimum −1 and period 360°. It crosses its midline upwards at x = 0°. Give a rule.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- State the amplitude, period and range of y = 4 sin(2x) + 1.3 marks
Use |A| = 4 and 360°/|B| = 180°.
Add and subtract 4 from the midline 1.
Amplitude 4; period 180°; −3 ≤ y ≤ 5. - Give y = cos x at x = 0°, 90°, 180°, 270° and 360°.2 marks
Use the x-coordinates of the unit-circle points.
1, 0, −1, 0, 1. - Sketch y = sin x for 0° ≤ x ≤ 360° on the supplied grid. Label the intercepts and turning points.4 marks
Evaluate sine at .
Plot the five key points and join with a smooth periodic curve.
Intercepts (0°,0), (180°,0), (360°,0); maximum (90°,1); minimum (270°,−1). - Describe how y = −2 cos x differs from y = 2 cos x.2 marks
Every y-value changes sign.
Both amplitudes are 2 and periods 360°.
It is reflected in the x-axis; amplitude and period stay the same. - Explain why sin(360° − x) = −sin x using the unit circle.3 marks
The two points have opposite vertical coordinates.
Sine is the vertical coordinate.
Reflection across the x-axis keeps x-coordinate but reverses y-coordinate. - A sine graph has maximum 3, minimum −1 and period 360°. It crosses its midline upwards at x = 0°. Give a rule.3 marks
Midline = (3 + (−1))/2 = 1.
Amplitude = (3 − (−1))/2 = 2.
An upward midline crossing at 0° matches sine.
y = 2 sin x + 1.
Curriculum reference & lesson scope
Australian Curriculum v9 and South Australian school course scope
Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.
Official curriculum source ↗