Graphs of the Sine and Cosine Function: Learn It 2

Period and Amplitude

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function.

general form of sine and cosine

The general forms of sinusoidal functions are

[latex]y = A\sin (Bx−C) + D[/latex]

and

[latex]y = A\cos (Bx−C) + D[/latex]

Determining the Period of Sinusoidal Functions

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.

In the general formula, B is related to the period by [latex]P=\frac{2π}{|B|}[/latex]. If [latex]|B| > 1[/latex], then the period is less than [latex]2π[/latex] and the function undergoes a horizontal compression, whereas if [latex]|B| < 1[/latex], then the period is greater than [latex]2π[/latex] and the function undergoes a horizontal stretch. For example, [latex]f(x) = \sin(x), B= 1[/latex], so the period is [latex]2π[/latex], which we knew. If [latex]f(x) =\sin (2x)[/latex], then [latex]B= 2[/latex], so the period is [latex]π[/latex] and the graph is compressed. If [latex]f(x) = \sin\left(\frac{x}{2} \right)[/latex], then [latex]B=\frac{1}{2}[/latex], so the period is [latex]4π[/latex] and the graph is stretched. Notice in the figure how period is indirectly related to [latex]|B|[/latex].

A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x/2) for one half of a period.

period of sinusoidal functions

For a sinusoidal function in the form [latex]y = A\sin (Bx−C) + D[/latex] or [latex]y = A\cos (Bx−C) + D[/latex], the period is [latex]\frac{2π}{|B|}[/latex].

Determine the period of the function [latex]f(x) = \sin\left(\frac{π}{6}x\right)[/latex].

Determine the period of the function [latex]g(x)=\cos\left(\frac{x}{3}\right)[/latex].

Determining Amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variable B relates to the period. Now let’s turn to the variable A so we can analyze how it is related to the amplitude, or greatest distance from rest. A represents the vertical stretch factor, and its absolute value [latex]|A|[/latex] is the amplitude. The local maxima will be a distance [latex]|A|[/latex] above the vertical midline of the graph, which is the line [latex]x = D[/latex]; because [latex]D = 0[/latex] in this case, the midline is the [latex]x[/latex]-axis. The local minima will be the same distance below the midline. If [latex]|A| > 1[/latex], the function is stretched. For example, the amplitude of [latex]f(x)=4\sin\left(x\right)[/latex] is twice the amplitude of [latex]f(x)=2\sin\left(x\right)[/latex]

If [latex]|A| < 1[/latex], the function is compressed. The graph below compares several sine functions with different amplitudes.

A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.

amplitude of sinusoidal functions

For a sinusoidal function in the form [latex]y = A\sin (Bx−C) + D[/latex] or [latex]y = A\cos (Bx−C) + D[/latex], the amplitude is |A|.

 

In addition, notice in the example that

[latex]|A|=\text{amplitude}=\frac{1}{2}|\text{maximum}−\text{minimum}|[/latex]

What is the amplitude of the sinusoidal function [latex]f(x)=−4\sin(x)[/latex]? Is the function stretched or compressed vertically?

What is the amplitude of the sinusoidal function [latex]f(x)=\frac{1}{2}\sin (x)[/latex]? Is the function stretched or compressed vertically?