Difficulty: Intermediate | Prerequisites: Unit circle, basic trigonometric functions (sin, cos), graphing parent functions
Sine and cosine are the building blocks for modelling anything that repeats: sound waves, tides, seasonal temperatures, rotating wheels. Before you can model real-world periodic behaviour, you need to understand how the parameters a, b, h, and k reshape the parent sine and cosine graphs. This topic sits right between "I know what sine and cosine are" and "I can write an equation for a Ferris wheel." If you are comfortable with the unit circle and can sketch y = sin(x) and y = cos(x) from memory, you are ready for this.
The parent sine and cosine curves both have an amplitude of 1 and a period of 2π. Changing a stretches or compresses the graph vertically (amplitude), changing b stretches or compresses it horizontally (period and frequency), and adding h or k shifts the whole curve left/right or up/down.
Amplitude
The distance from the midline to the peak (or trough) of a sine or cosine curve. For y = a sin(bx), the amplitude is |a|.
In simple terms, amplitude tells you how tall the wave is from its centre line.
Period
The horizontal length of one complete cycle. Calculated as 2π / |b|.
Think of it as: how far along the x-axis you have to travel before the pattern starts repeating.
Frequency
The number of complete cycles that occur in one standard interval (0 to 2π). Calculated as |b| / (2π), or equivalently 1 / period.
In simple terms, frequency is how many times the wave repeats in a given stretch. Higher frequency means more cycles packed into the same space.
Translation
A vertical or horizontal shift of the entire graph. Vertical translation adds k outside the function; horizontal translation subtracts h inside the argument.
Think of it as sliding the whole curve up/down or left/right without changing its shape.
Phase shift
Another name for a horizontal translation. The value h in y = sin(x − h) shifts the curve h units to the right.
In simple terms, a phase shift just moves the starting point of the wave along the x-axis.
Midline
The horizontal line y = k around which the curve oscillates. For the parent functions, the midline is y = 0.
Think of it as the resting position of the wave, halfway between its highest and lowest points.
y = sin(x) starts at the origin (0, 0), rises to a peak of 1 at π/2, returns to 0 at π, dips to −1 at 3π/2, and completes the cycle at 2π.
y = cos(x) starts at its peak (0, 1), drops to 0 at π/2, reaches −1 at π, returns to 0 at 3π/2, and completes the cycle at 2π.
Both parent functions have amplitude = 1, period = 2π, and frequency = 1/(2π).
In the general form y = a sin(bx) or y = a cos(bx), both a and b equal 1 for the parent curves.
|a| > 1 stretches the graph vertically, making the peaks and troughs farther from the midline.
0 < |a| < 1 compresses the graph vertically, flattening the wave.
If a is negative, the graph is reflected across the x-axis (the curve is inverted, starting at a trough instead of a peak for cosine, or dipping first instead of rising for sine).
The period becomes 2π / |b|.
|b| > 1 compresses the wave horizontally: cycles happen more often.
0 < |b| < 1 stretches the wave horizontally: cycles happen less often.
The frequency becomes |b| / (2π), which is simply the reciprocal of the period.
y = sin(x) + k shifts the entire curve up by k units (or down, if k is negative).
The midline moves from y = 0 to y = k.
The shape, period, and amplitude are unchanged.
y = sin(x − h) shifts the curve h units to the right. If h is negative, the shift goes left.
A horizontal translation is called a phase shift because adding a multiple of 2π to h produces an identical graph. For this reason, h is usually kept between 0 and the period.
The shape, period, and amplitude are unchanged.
General sine model: y = a sin[b(x − h)] + k
General cosine model: y = a cos[b(x − h)] + k
Period: Period = 2π / |b|
Frequency: Frequency = |b| / (2π) = 1 / Period
Solving for b from a known period T: b = 2π / T
Solving for b from a known frequency f: b = 2πf
Sound is a pressure wave, and musical notes correspond to specific frequencies. When you see "440 Hz" on a tuner, that is 440 complete sine-wave cycles per second. Understanding how b controls frequency is exactly how audio engineers and instrument designers think about pitch.
Students often think a larger b means a longer period. It is the opposite: larger b compresses the wave, making the period shorter.
Students confuse amplitude with the total vertical span. The amplitude is half the distance from peak to trough, not the full distance.
Students sometimes apply the phase shift in the wrong direction. In y = sin(x − h), positive h shifts right, not left. The minus sign inside the bracket is part of the standard form.
Students forget that a negative a flips the graph. A negative amplitude does not exist; the amplitude is always |a|, but the sign of a determines whether the curve is inverted.
⚠️ You will almost certainly be asked to identify amplitude, period, and phase shift from a general equation. Know the formulas cold.
⚠️ Expect at least one question where you must distinguish between sine and cosine based on the initial value (does the function start at the midline or at a peak/trough?).
⚠️ Converting between frequency and period (and finding b from either) is a routine exam step. Practise both directions.
True or False: The amplitude of y = −3 sin(x) is −3.
The period of y = cos(4x) is __________.
True or False: y = sin(x − π/2) shifts the graph π/2 units to the left.
If the frequency of a wave is 60 Hz, the value of b is __________.
True or False: Changing k in y = sin(x) + k affects the period.
Answers: 1. False (amplitude is 3). 2. 2π/4 = π/2. 3. False (shifts right). 4. b = 2π(60) = 120π. 5. False (only shifts the midline).
Q: What is the amplitude, period, and midline of y = 5 sin(2x) + 3?
A: Amplitude = 5, period = 2π/2 = π, midline y = 3.
Q: Write the equation of a cosine function with amplitude 4, period 6π, and no vertical or horizontal shift.
A: Period = 2π/b, so b = 2π/(6π) = 1/3. The equation is y = 4 cos(x/3).
Q: A sine curve has been shifted 3 units up and π/4 units to the right, with amplitude 2 and period π. Write its equation.
A: b = 2π/π = 2. The equation is y = 2 sin[2(x − π/4)] + 3.
Q: Explain why y = sin(x) and y = cos(x − π/2) produce the same graph.
A: Cosine shifted right by π/2 starts at 0 instead of at its peak, which matches the starting behaviour of sine. The co-function identity confirms sin(x) = cos(x − π/2).
This material connects directly to the next topic, Modeling with Periodic Functions, where you will use these transformations to write equations for real-world scenarios (springs, sound waves, Ferris wheels). It also ties back to the unit circle: the coordinates (cos θ, sin θ) on the circle are the same values plotted as functions of θ. Later in calculus, the derivatives of sine and cosine rely on understanding amplitude and period.
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