The term sinusoidal is used to describe a curve, referred to as a sine wave or a sinusoid, that exhibits smooth, periodic oscillation. It is named based on the function y=sin(x). Sinusoids occur often in math, physics, engineering, signal processing and many other areas.
Graph of y=sin(x)
Below are some properties of the sine function:
- Domain: -∞<x<∞
- Range: -1≤y≤1
- Period: 2π – the pattern of the graph repeats in intervals of 2π
- Amplitude: 1 – the sine graph is centered at the x-axis. The amplitude is the distance between the line around which the sine function is centered (referred to here as the midline) and one of its maxima or minima
- Zeros: πn – the sine graph has zeros at every integer multiple of π
- sin(-x)=-sin(x) – the graph of sine is odd, meaning that it is symmetric about the origin
Most applications cannot be modeled using y=sin(x), and require modification. The equation below is the generalized form of the sine function, and can be used to model sinusoidal functions.
y = A·sin(B(x-C)) + D
where A, B, C, and D are constants such that:
- is the period
- |A| is the amplitude
- C is the horizontal shift, also known as the phase shift. If C is positive, the graph shifts right; if it is negative, the graph shifts left
- D is the vertical shift. If D is positive, the graph shifts up; if it is negative the graph shifts down
- the sinusoid is centered at y = D
1. Graph y = 3sin(2x)
- Amplitude: |A| = |3| = 3
- C = 0, so there is no phase shift
- D = 0, so there is no vertical shift
Two periods of the graph are shown below. The graph of y = sin(x) is also shown as a reference.
2. Graph y = 2sin(x - ) + 3.
- Amplitude: |A| = |2| = 2
- C = , so the graph shifts right
- D = 3, so the graph shifts up 3
The graph are shown below.
Equation of a sinusoidal curve
Given the graph of a sinusoidal function, we can write its equation in the form y = A·sin(B(x - C)) + D using the following steps.
- D: To find D, take the average of a local maximum and minimum of the sinusoid. y=D is the "midline," or the line around which the sinusoid is centered.
- A: To find A, find the perpendicular distance between the midline and either a local maximum or minimum of the sinusoid. For example, y=sin(x) has a maximum at (, 1), and is centered about y=0. Subtracting their y-values yields A = 1 - 0 = 1.
- B: Examine the graph to determine its period. Choose an easily identifiable point on the sinusoid, such as a local maximum or minimum, and determine the horizontal distance before the graph repeats itself. This is the period of the graph. B=.
- C: To find C, graph the line y=D. Look at the first points left and right of the y-axis where the sinusoid intersects y=D. Choose the point of intersection that precedes a local maximum of the sinusoid (the function is increasing immediately to the right of the point); The x-value of this point is C.
Write an equation for the sinusoidal graph below.
- The maximum value of the graph is 3 and the minimum value is -1, so the equation of the midline is,
- The sinusoid has maximum at y = 3, and D = 1, so
A = 3 - 1 = 2
- There is a maximum at x=. The next maximum after that is at x= so the period is .
- The first point at which the sinusoid intersects the line y=1 that precedes a local maximum is .
Substituting all of these into the generalized form of the sine function:
Because of the periodic nature of a sinusoid, the equation for a sinusoidal curve is not unique. We could have found different points for C, such as (, 1) or (, 1), and their equations,
would result in the same curve.