Sine and Cosine Graphs Explained
Every sine and cosine graph you will be asked about is
y = a·sin(b(x - c)) + d. Four numbers, four independent
effects. Learn what each one does on its own and any combination becomes
readable.
The basic wave
y = sin(x) oscillates between -1 and 1, completes one full
cycle every 2π, and passes through the origin heading upward.
y = cos(x) is the same wave started a quarter-cycle earlier: it
begins at its maximum of 1 when x = 0.
Use the π button. The toolbar above the graph has a
π button that switches the x-axis to multiples of π and sets
a window suited to trigonometry. Labels reading π/2 and 3π/2 are far
easier to work with than 1.57 and 4.71.
a: amplitude
a stretches the wave vertically. The amplitude is
|a|, the distance from the midline to a peak, so the curve runs
between -|a| and +|a|.
A negative a also flips the curve upside down.
y = -sin(x) has amplitude 1 but starts by heading downward.
Amplitude changes nothing horizontal. The zeros stay exactly where they were.
b: period
b squeezes the wave horizontally. The period — the
x-distance for one complete cycle — is:
period = 2π / |b|
Larger b means more cycles packed into the same space.
y = sin(2x) has period π and fits two cycles where
the basic curve fits one. y = sin(x/2) has period
4π and is stretched out.
c: phase shift
c slides the wave sideways. In the form
y = sin(b(x - c)), the shift is c to the right.
Here is where almost everyone slips. If the equation is written
y = sin(2x - π), the phase shift is not π. You
have to factor b out first:
sin(2x - π) = sin(2(x - π/2))
So the shift is π/2, not π. The general rule for
sin(bx - k) is that the shift equals k/b.
d: midline
The easiest of the four. d moves the whole wave up or down, and
the horizontal line y = d becomes the new midline. The curve
then runs from d - |a| to d + |a|.
Putting them together
Take y = 2sin(3(x - π/6)) + 1 and read it off:
| Part | Value | Effect |
|---|---|---|
a = 2 | Amplitude 2 | Runs from 1 − 2 = −1 up to 1 + 2 = 3 |
b = 3 | Period 2π/3 | Three cycles in the usual space for one |
c = π/6 | Shift right by π/6 | Already factored, so read it directly |
d = 1 | Midline y = 1 | Whole wave raised by 1 |
Learn it with sliders. Type
a*sin(b*(x - c)) + d into the calculator and it offers a
slider for each letter. Change one at a time and watch which feature of
the curve responds. Ten minutes of that beats memorising the table.
Open the calculator.
Reading an equation off a graph
Given a picture of a wave, recover the equation in four steps.
- Midline. Average the maximum and minimum:
d = (max + min)/2. - Amplitude. Half the distance between them:
a = (max - min)/2. - Period. Measure one complete cycle, peak to peak, then
b = 2π/period. - Phase shift. Find where the curve crosses the midline
going upward. For a pure sine that happens at
x = 0, so however far along it happens here isc.
If the curve starts at a peak rather than on the midline, use cosine instead. You will get the same graph with a simpler phase shift, and either answer is correct.
Questions
What is the formula for the period of a sine graph?
Period = 2π divided by |b|, where b is the coefficient of x. For y = sin(3x) the period is 2π/3, so the wave completes three full cycles in the space the basic sine curve takes for one.
What is the difference between phase shift and horizontal shift?
They are the same movement described differently. The catch is that you must factor b out first: in y = sin(2x − π), the shift is π/2, not π, because the expression factors as sin(2(x − π/2)).
Can amplitude be negative?
Amplitude is defined as a distance, so it is always positive and equals |a|. A negative value of a does something real, though: it reflects the curve vertically, so it starts by going down instead of up.
How are sine and cosine related?
Cosine is sine shifted left by π/2. They are the same wave started at a different point, which is why cos(x) = sin(x + π/2) holds for every x.
How do I find the equation of a sine graph from a picture?
Read the midline for d, take half the distance from peak to trough for a, measure one full cycle and use b = 2π/period, then compare where the curve crosses the midline going upward against where sin(x) does, which gives the phase shift.