Graph sine and cosine with amplitude and period
Sketch translated sine and cosine curves by identifying their midline, amplitude, period and five key points on the stated domain.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the midline of y = a sin bx + c?
What you will learn
- Find amplitude, period and vertical translation in sine and cosine functions
- Sketch a full cycle in degrees or radians using quarter-period points
- Describe how a negative coefficient or a changed frequency alters a graph
Read the shape from y = a sin bx + c or y = a cos bx + c
The midline is y = c. The amplitude is a, the distance from the midline to a maximum or minimum; in 0606 the stated graph forms use a positive integer. The range is c − a ≤ y ≤ c + a.
The period is 360°/|b| in degrees or 2π/|b| in radians for non-zero b. The coefficient b changes the horizontal scale; c shifts the curve up or down.
Use five points to sketch one cycle
For sine, one cycle starts on the midline, reaches a maximum after one quarter-period, returns to the midline halfway through, reaches a minimum after three quarters and returns to the midline at the end.
For cosine, one cycle starts at a maximum when a is positive, then follows the same quarter-period spacing. Mark the stated domain and use points within it; do not assume the graph must start at zero.
For example, y = sin x has amplitude 1 and period 2π, while y = 3 sin 2x has amplitude 3 and period π. Doubling b halves the period; multiplying by 3 stretches the graph vertically. Sine and cosine curves with the same a and b have the same amplitude and period but start at different points in the cycle. In degrees or radians, keep all x-values in the same unit as the question.
Pause and explain
If y = 4 cos(2x) − 3 uses radians, what is its range?
Worked example
Sketch y = 3 sin 2x + 1 for 0° ≤ x ≤ 180°. State its amplitude, period, midline, range and key points.
Show the worked solution
- Amplitude is 3, midline is y = 1, and the period is 360°/2 = 180°.
- The range is −2 ≤ y ≤ 4. A quarter-period is 45°.
- The sine curve passes through (0°,1), (45°,4), (90°,1), (135°,−2) and (180°,1). Join smoothly.
Answer Amplitude 3; period 180°; midline y = 1; range [−2, 4]; key points (0°,1), (45°,4), (90°,1), (135°,−2), (180°,1).
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For y = 2 cos(πx/3) − 1 on 0 ≤ x ≤ 6, find the amplitude, period, range and five key points.
Give me a hint
The period is 2π divided by the coefficient of x; space the points one quarter-period apart.
Compare my reasoning
- Amplitude is 2, midline y = −1, period is 2π/(π/3) = 6, and range is [−3,1].
- A quarter-period is 1.5. Cosine starts at its maximum, so the x-values are 0, 1.5, 3, 4.5 and 6.
- The corresponding y-values are 1, −1, −3, −1 and 1.
Amplitude 2; period 6; range [−3,1]; points (0,1), (1.5,−1), (3,−3), (4.5,−1), (6,1).
Look for these in your work
- Amplitude, shift and period correct
- Quarter-period spacing correct
- Cosine starting maximum used
Sketch y = 3 sin x + 2 for 0 ≤ x ≤ 2π. State its range and key points.
Give me a hint
Use the five sine landmarks one quarter-period apart.
Compare my reasoning
- Amplitude is 3, midline is y = 2 and period is 2π.
- At x = 0, π/2, π, 3π/2, 2π, sine is 0, 1, 0, −1, 0.
- Apply y = 3 sin x + 2 to get 2, 5, 2, −1, 2.
Range [−1,5]; key points (0,2), (π/2,5), (π,2), (3π/2,−1), (2π,2).
Look for these in your work
- Amplitude and midline identified
- Period and key x-values correct
- Range matches extrema
A daily temperature model is T = 18 + 4 cos(πt/12), where t is hours after midnight. Give the midline, amplitude, period and times of the maximum and minimum for 0 ≤ t ≤ 24.
Give me a hint
The cosine is at a maximum at t = 0 and one full period later; halfway through the period it reaches its minimum.
Compare my reasoning
- The midline temperature is 18 and the amplitude is 4, so the range is 14 to 22.
- The period is 2π/(π/12) = 24 hours.
- Maximum 22 occurs at t = 0 and 24; minimum 14 occurs at t = 12.
Midline 18; amplitude 4; period 24 hours; maximum 22 at t = 0 and 24, minimum 14 at t = 12.
Look for these in your work
- Model parameters interpreted
- Period converted to hours
- Extrema match the cosine cycle
Common mistakes
- Calling c the amplitude instead of the vertical shift
- Using 360b as the period rather than 360°/|b|
- Confusing degrees and radians
- Drawing straight segments instead of a smooth periodic curve
What can you explain now?
What is the period of y = 2 cos(3x) when x is measured in radians?
Compare with the explanation
2π/3
The radians period is 2π divided by |b|, so 2π/3.
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Make it stick
How do a, b and c determine amplitude, period, midline and range in a sine or cosine graph?
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