| Properties of Sine and Cosine Graphs |
Certain mathematical functions repeat themselves across their domain.
The y-values occur within a precise range over and over again.
Such functions are called periodic or cyclic functions.
Here are the graphs of two periodic functions.
As we see, both graphs repeat their y-values every 4 units of the x-axis.
We say that the period of these graphs is 4 units.
The range of f (x) on the left is 
, and
the range of g (x) on the right is 
.
| A periodic function f has the property that the y-values are the same for x1 and x2 (in the domain). The interval x2 – x1 is called the period of function f. |
All the trigonometric functions are periodic functions.
| intro | basic sine | transformed sine | practice | solutions |
Properties of Sine and Cosine Graphs:
The Basic Sine Curve: f (x) = sin x.
Since radian measure represents length as well as rotation (angles),
we express our x-values in radians and our y-values in sin x values.
Using this approach, we can graph trig functions.
From the x and y values of the unit circle we know:
| sin 0 = 0 |  |
 |
 |
Here, we see two cycles of f (x) = sin x.
This pattern will repeat to the right & left indefinitely.
As you can see, the domain is R, the range is [ – 1, 1].
There are infinitely many zeros, each of which is located at the midpoint
between the nearest maximum and minimum.
The properties of the cyclical trig graphs still include all the elements of the former list, (ie: domain, range, max, min, increasing, decreasing and signs) but it also includes:
amplitude, period, frequency, phase shift and vertical translation.
| intro | basic sine | transformed sine | practice | solutions |
.
The Transformed Sine Function: f (x) = a sin b( x – h ) + k
amplitude: the amplitude of a sine or cosine curve is the measure of the distance it
moves above or below its horizontal axis.
It is found by taking 1/2(ymax - ymin). On the basic curve above, the amplitude is 1.
| amplitude = | a | | a > 0 curve moves up (note) | a < 0 curve moves down (note) |
period : the period of a sine or cosine curve is the domain interval of a single cycle.
On the basic curve above, the period is 
, since the y-values will have attained
every value between – 1 and 1.
period = | | |
if b < 0, same effect as a < 0. (note) |
frequency: the frequency of a sine or cosine curve is defined as the reciprocal of the period.
On the basic curve above, the frequency is 
.
frequency = 
.
phase shift: the phase shift of a sine or cosine or cosine curve is a fancy name
for the horizontal translation effected by h.
phase shift = h.
vertical translation : the vertical translation is the up or down movement of the horizontal axis of the curve. As usual, the vertical translation is k.
| vertical translation = k | maximum: k + | a | | minimum: k – | a | |
Now let's learn a simple way to graph a transformed sine curve.
As we see from the basic sine curve above, the curve is divided into 4 equal parts:
from zero to max, from max to zero, from zero to min, and from min to zero.
This is true of all sinusoidal curves.
So, we establish our start point (h, k), add the period to h, find the end point 
then, using midpoints, we divide this interval into 4 equal parts to locate the maximum, minimum and on-axis points.
Note1: if either a or b are negative ( < 0 ), the curve moves down from it's starting point to the min value at the beginning of a cycle rather than vice versa.
So, had we graphed y = – sin x or y = sin ( – x ), the curve would have initially moved down
to 
instead of rising to 
.
If both a and b are negative, the curve behaves as if both are positive.
In a way, they cancel each other out.
Note2: For cosine curves, a and b work differently since the cosine of an angle in the 4th quad is positive, so if b is negative there's no change in the sign of the cosine.
With cosine curves, the sign of a determines if the curve starts at max or min, since cosine curves start at an extreme value rather than on their horizontal axis like sine curves.
Note3: A cosine curve is just a sine curve hauled backwards a quarter of the period.
Note that the Blue Cosine Curve starts at (0, 1), the maximum,
instead of (0, 0) the "on-axis value" like the red Sine curve.
.
.
Example
List all function properties for:
a)  |
b)  | ||
| initial direction: a < 0 moves down to min | initial direction: a > 0 starts at maximum | ||
| amplitude: | a | = 5 | amplitude: | a | = 4 | ||
period:  |
period:  | ||
phase shift:  |
phase shift:  | ||
start:  (starts on axis) |
start:  (starts at max) | ||
end:  |
end:  | ||
¼ cycle points:  |
¼ cycle points:  | ||
| vertical translation: k = 3 | vertical translation: k = – 1 | ||
| min: k – a = – 2 | max: k + a = 8 | min: k – a = – 5 | max: k + a = 3 |
.
| intro | basic sine | transformed sine | practice | solutions |
Put these function rules in standard form f (x) = a sin b(x – h) + k
Copy the list of properties from the table and find the values for these functions.
| intro | basic sine | transformed sine | practice | solutions |
1/  |
2/  | ||
 |
 | ||
| initial direction: a > 0, b < 0 down to min | initial direction: a < 0 starts at min | ||
| amplitude: | a | = 14 | amplitude: | a | = 3 | ||
period:  |
period:  | ||
phase shift:  |
phase shift:  | ||
start:  (starts on axis) |
start:  (starts at min) | ||
end:  |
end:  | ||
¼ cycle points:  |
¼ cycle points:  | ||
| vertical translation: k = – 12 | vertical translation: k = 7 | ||
| min: k – a = – 26 | max: k + a = 2 | min: k – a = 4 | max: k + a = 10 |
For more on this topic, see the Trig MathRoom Lesson Graphing Sine and Cosine Curves.
| intro | basic sine | transformed sine | practice | solutions |
(all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).