1. Lesson 3

1.2. Explore

Mathematics 30-2 Module 6

Module 6: Sinusoidal Functions

 

Explore

 

In Discover you explored how a, b, c, and d are related to characteristics of the graph y = a sin b(x − c) + d with the angle measured in radians. You may have found the following:

  • The amplitude of the graph is determined by the parameter a of the function.
  • The parameter b is related to the period by the equation , where P represents the period and the angle is measured in degrees. Notice that a large b-value results in a shorter period, and a small b-value results in a longer period.

This diagram shows the amplitude of a sine function is equal to a and the period of a sine function is equal to 360 degrees divided by b.

 

The parameters c and d do not change the shape of the graph, but they do move the graph vertically and horizontally.

 

When the graphs are moved up or down, three characteristics are affected: the midline, the maximum, and the minimum.

  • The midline will occur at y = d.
  • The maximum value = d + a.
  • The minimum value = d − a.

In Discover you noticed that changing the parameter c results in the graph being shifted c units horizontally.

 

This diagram shows the midline of a sinusoidal equation occurs at d and the horizontal translation is equal to c.



caution

Inserting a positive c-value into y = a sin b(x − c) + d makes the c term appear negative.

  • c = 5 in y = a sin b(x − 5) + d is a translation of 5 units to the right of y = sin x.
  • c = −5 in , which is equal to y = a sin b(x + 5) + d, is a translation of 5 units to the left of y = sin x.

textbook

Read parts a. and b. of “Example 2” on page 519 of your textbook. The solutions are provided on page 520. This example shows how an equation of the form y = a sin b(x − c) + d can be interpreted.

 

 

Self-Check 1

 

Complete questions 1, 2, 5.a., 7.a., 8.a., and 9.a. on page 528 of your textbook. Answers