5.3 – Trigonometric Graphs
5.3 Video 1 of 3:Sine and Cosine curves
5.3 Video 2 of 3:Amplitude, Period and Phase Shift
5.3 Video 3 of 3:Trig Graph Examples
Trigonometric Graphs (Sin and Cos)
Viewing the graph of a function can give insight to its behavior. This section will examine the graphs of the trigonometric functions.
We first want to note that, as we are dealing with a unit circle, when circumnavigates this circle, the sine and cosine values repeat themselves. This is known as periodicity. We say that sine and cosine are periodic functions. This can be stated mathematically:
|Periodic Properties of Sine and Cosine|
|Sine and Cosine both have period , thus:
Because the functions repeat themselves every , we can look at one period to see the behavior of the graph. We will do this by taking as the horizontal axis of our graph, and the function as the vertical. First, let us look at the function , and begin by listing out the known values:
Using the same method, we can find the points of , graph them, connecting them with a smooth curve through one period:
The graphs above are just one period of each of the functions. They continue forever in each direction in a similar fashion. Let us expand these functions further:
The graph of :
And the graph of :
Trigonometric graphs can be shifted up, down, left or right by quick manipulations. To shift the graph of up by 3 units, 3 is simply added to the function. That is, the function is the graph of shifted up by 3 units:
The graph of can also be “stretched” vertically by multiplying by a constant number. Shown below are a few multiples of the Sine function:
And of course, these functions can be multiplied by a negative constant number. This will reflect the function across the horizontal axis. Here are a few graphs of the Cosine function and the Cosine function multiplied by various negative numbers:
Trigonometric functions can also have their periods altered. Below is the graph of and on a single set of axes:
You might notice that the period of the function is smaller than the period of . In fact, there are two periods of in the same interval that holds one period of .
Below is graphed one period of an assortment of different periodic Sine functions:
Another way to manipulate these graphs is a horizontal shift. This can be accomplished by adding or subtracting a number within the trigonometric function itself. The graph of is the graph of shifted to the right by 1 unit. the graph of is the graph of shifted to the left by 1 unit.
These graph manipulations can be put together to describe the shape of many Sine and Cosine functions:
|Sin and Cosine Graphs|
|The sine and cosine curves
have an amplitude of , period of , are horizontally shifted by units, and vertically shifted by units.
One period is graphed between the interval
|Find the amplitude, period, horizontal and vertical shifts and then graph the function|
|This graph will have an amplitude of 3, a period of , a horizontal shift of 3 and a vertical shift of 2. One period is graphed in the interval :
And to show this graph over multiple periods, it is simply extended to the left and the right using the same periodic length: