Sine and Cosine Graphs
Understand the shape, amplitude, period, phase shift and vertical shift of sine and cosine functions, and how transformations affect their graphs.
The General Form
The general sine and cosine functions are written as:
y = a sin(bx − c) + d
y = a cos(bx − c) + d
a — Amplitude
The height from the centre line to the peak. Amplitude = |a|. If a < 0, the graph is reflected in the x-axis.
b — Period modifier
The period (length of one cycle) = 2π|b|. Larger b means more cycles, shorter period.
c — Phase shift
Horizontal shift = cb. Positive c shifts the graph to the right.
d — Vertical shift
Moves the entire graph up (positive d) or down (negative d). The centre line becomes y = d.
Basic Sine and Cosine Curves
The basic y = sin x and y = cos x curves both have amplitude 1 and period 2π. The key difference is their starting position:
y = sin x
Starts at (0, 0), rises to a peak at π/2, falls through zero at π, dips to a trough at 3π/2.
y = cos x
Starts at (0, 1), falls through zero at π/2, reaches a trough at π, rises back through zero at 3π/2.
How Transformations Affect the Graph
| Transformation | Example | Effect |
|---|---|---|
| Amplitude change | y = 3 sin x | Stretches vertically; amplitude becomes 3 |
| Reflection | y = −cos x | Flipped upside down (reflected in x-axis) |
| Period change | y = sin(2x) | Period = 2π/2 = π (twice as many cycles) |
| Phase shift | y = sin(x − π/3) | Shifts right by π/3 units |
| Vertical shift | y = cos x + 2 | Shifts up by 2 units; centre line at y = 2 |
Key Vocabulary
Amplitude
The maximum displacement from the centre line of the graph. Always positive.
Period
The horizontal length of one complete cycle of the wave. Default is 2π.
Phase Shift
A horizontal translation of the graph, left or right along the x-axis.
Centre Line
The horizontal line y = d about which the wave oscillates. Default is y = 0.
Worked Examples
Find the amplitude and period of y = 3 sin(2x).
Step 1: Amplitude = |a| = |3| = 3.
Step 2: Period = 2π/|b| = 2π/2 = π.
Answer: Amplitude = 3, Period = π.
Describe the transformations of y = −2 cos(x − π/4) + 1.
Step 1: a = −2: Amplitude = 2, reflected in the x-axis.
Step 2: b = 1: Period = 2π/1 = 2π (no change to period).
Step 3: Phase shift = π/4 to the right.
Answer: Amplitude 2, reflected, period 2π, shifted right π/4, shifted up 1.
Write the equation of a sine curve with amplitude 4, period π, and shifted up 3.
Step 1: Amplitude = 4, so a = 4.
Step 2: Period = π = 2π/b, so b = 2.
Step 3: Vertical shift = 3, so d = 3.
Answer: y = 4 sin(2x) + 3
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the period of y = sin(3x)?
Question 2
What is the amplitude of y = −5 cos(x)?
Question 3
The graph of y = sin(x − π/2) is the same as which function?
Question 4
What is the range of y = 2 sin(x) + 3?
Question 5
A cosine graph has amplitude 3 and period 4π. What is the value of b?
Key Concepts Summary
- ●Amplitude = |a|, the maximum displacement from the centre line.
- ●Period = 2π/|b|, the horizontal length of one complete cycle.
- ●Phase shift = c/b, horizontal translation of the graph.
- ●Vertical shift = d, moves the centre line to y = d.
- ●The range of the function is [d − |a|, d + |a|].