1. Lesson 3

1.9. Lesson 3 Summary

Mathematics 30-2 Module 6

Module 6: Sinusoidal Functions

 

Lesson 3 Summary

 

A sine function and cosine function can be written in the standard forms y = a sin b(x − c) + d and y = a cos b(x − c) + d, where a, b, c, and d represent real numbers and are called parameters. Parameters can be changed so that the function can be used for many different applications.

 

The characteristics of sinusoidal functions can be summarized as follows.

 

The value of a is the amplitude:

 

 

 

The value of d determines the midline and helps find the maximum and minimum values.

  • equation of the midline is y = d
  • maximum value = a + d
  • minimum value = a − d

The value of b is the number of cycles in 360° or 2π. The period can be calculated using this formula:

 

           

 

The value of c designates the horizontal translation.

  • A positive c moves the y = sin x or y = cos x function c units to the right.
  • A negative c moves the y = sin x or y = cos x function c units to the left.

 

In this illustration the amplitude, period, midline, and phase horizontal translation are labelled on a sinusoidal graph. The illustration also shows that the amplitude equals a, the period equals 2 pi divided by b, the midline equals d, and the horizontal translation equals c.

In this diagram, the period is written as , which means this graph is in radians.
If the graph is in degrees, use

 

In Lesson 4 you will investigate how sinusoidal functions can be used to model some real-world scenarios.