1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 5
Trigonometric Functions
5.5 Part 4 Graphs of Sine and Cosine Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Understand the graph of y = sin x.
Understand the graph of y = cos x.
Graph variations of y = sin x.
Graph variations of y = cos x.
Use vertical shifts of sine and cosine curves.
Model periodic behavior.

3. Vertical Shifts
The vertical movement of a graph up or down from its original position is called a “vertical shift”.
A vertical shift of a circular function occurs when a term is added to, or subtracted from, the basic sine or cosine function.
When a positive number is added to the basic sine or cosine function the vertical shift is up.
When a positive number is subtracted from the basic sine or cosine function the vertical shift is down.
In both these cases the effect is to move the “x-axis” from y = 0 to y = the amount of the vertical shift.

4. Vertical Shifts
The vertical shift is caused by the constant d in the equations
y = d + a sin(bx – c) and y = d + a cos(bx – c).
The shift is d units up for d > 0 and d units down for d < 0.
In other words, the graph oscillates about the horizontal line y = d instead of about the x-axis.
The maximum value of y is d + |a|.
The minimum value of y is d – |a|.

5. Example: Vertical Shift in the Sine Function
Compared with y = sin x, the graph y = 1 + sin x will be moved 1 unit up.
The amplitude will still be “1” and the period will still be 2π.
There will be no phase shift

6. Example: Sketch the graph of y = 1 + sin x
The primary period of this function will occur when:
0 ≤ x ≤ 2π
Divide this interval into quarter points:
Assign pattern of sine values (0, 1, 0, -1, 0), with “1” added to each, the five key points are:
Connect the points with a smooth curve:

7. Example: Sketch the graph of y = 1 + sin x
Points to be connected with smooth curve:

8. Example:
Graph one period of the function y = 2cosx + 1 (same as y = 1 + 2cos x).
Identify the amplitude, period, phase shift, and vertical shift.
y = acos(bx – c) + d or y = d + acos(bx – c)
y = 2cosx + 1
amplitude: |a| = |2| = 2 period: 2π/b = 2π/1 = 2π
phase shift: none vertical shift: d = 1 unit up

9. Example: y = 2cosx + 1
Find the values of x for the five key points.

10. Example: y = 2cosx + 1
Find the values of y for the five key points.

11. Example: y = 2cosx + 1
3. (cont) Find the values of y for the five key points.

12. Example: y = 2cosx + 1
Connect the five
key points with
a smooth curve
and graph one
complete cycle of
the given function.

13. Your Turn: Graph 2  2 sin 3x
Sketch the graph of 2  2 sin 3x.

14. Your Turn:
Sketch the graph y = 1 + 2 sin (4x + ).