This document discusses how to translate and write equations for graphs of sine and cosine functions by modifying their amplitude, period, vertical translation, and phase shift. It provides examples of writing equations from graphs based on identifying these parameters. Key aspects covered include how horizontal and vertical shifts affect the graph, defining phase shift, and techniques for determining the equation components like amplitude, period, and phase shift from a graph.

1. 6.4 Translating Graphs of Sine
and Cosine
Chapter 6 Circular Functions and Their Graphs

2. Concepts and Objectives
 Be able to identify how the graphs of the sine and cosine
change due to changes in
 Amplitude
 Period
 Vertical translation
 Phase shift

3. Translating Sine and Cosine
 We have seen what the graph of y = a sin bx looks like.
Next, we can shift the graph vertically and/or
horizontally.
 The full form of the sine function is
 d affects the vertical position of the graph. A positive
d shifts the graph d units up, and a negative d shifts
the graph d units down.
 c shifts the graph horizontally. x + c shifts the graph
c units to the left, and x − c shifts the graph c units
to the right.
 siny d a b x c  

4. Translating Sine and Cosine
 With circular functions, a horizontal translation is called
a phase shift. The phase shift is the absolute value of c.
 To sketch the translated graph, you can either divide the
interval into four parts (eight parts for two periods) and
chart the values as before, or you can sketch the
stretched/compressed parent graph and translate it
according to c and d.
 The second method is probably the easiest to do once
you are comfortable with the basic graphs.

5. Graphing Sine and Cosine
 Example: Graph over one period. 
 
 

 3cos
4
y x

6. Graphing Sine and Cosine
 Example: Graph over one period. 
 
 

 3cos
4
y x

   3, 1, to the left, 0
4
a b c d

7. Graphing Sine and Cosine
 Example: Graph over one period. 
 
 

 3cos
4
y x

   3, 1, to the left, 0
4
a b c d

8. Graphing Sine and Cosine
 Example: Graph over one period. 
 
 

 3cos
4
y x

   3, 1, to the left, 0
4
a b c d

9. Graphing Sine and Cosine
 Example: Graph over one period. 
 
 

 3cos
4
y x

   3, 1, to the left, 0
4
a b c d

10. Graphing Sine and Cosine
 Example: Graph over one period. 
 
 

 3cos
4
y x

   3, 1, to the left, 0
4
a b c d

11. Graphing Sine and Cosine
 Example: Graph over two periods.   1 2sin 4y x

12. Graphing Sine and Cosine
 Example: Graph over two periods.
To find the value of b, we will have to factor out the 4 in
front of the x:
   1 2sin 4y x
 
 
 

  1 2sin4
4
y x
2, 4, 1, to the left
4
a b d c

   
  
 
2 2
Period:
4 2b

13. Graphing Sine and Cosine
 Example: Graph over two periods.   1 2sin 4y x
 
 
 

  1 2sin4
4
y x

   2, 4, 1,
4
a b c d
Period:
2

14. Writing a Function From a Graph
 To write a function from the graph:
 Identify the vertical shift (d) and draw a midline.
 Count from the midline to a peak – this is the
amplitude (a).
 Identify the beginning and end of one period. Find
this length. To calculate b, re-write as ,
and solve for b.
 If the graph crosses the y-axis at the midline, use the
sine function; if it crosses the y-axis at a peak, use
cosine. In either case, c = 0.


2
period
b

15. Writing a Function From a Graph
 If it crosses somewhere else, decide whether to use
sine or cosine.
 If you are using sine, c is the distance from the
y-axis to where the graph crosses the midline.
 If you are using cosine, c is the distance from the
y-axis to a peak.
 Determine whether a should be positive or negative
by comparing your graph to the parent graph.
 Write your function:
or siny d a b x c    cosy d a b x c  

16. Writing Equations From Graphs
 Example: Write an equation for the graph below.

17. Writing Equations From Graphs
 Example: Write an equation for the graph below.
1. Find the middle of the
graph. This tells us that
d = 1 and a = 1.
2. Shift the graph so that
the middle is on the
x-axis.
3. Since the graph goes
through the origin, we
will use sine.

18. Writing Equations From Graphs
 Example: Write an equation for the graph below.
4. Since the graph goes
through the origin, we
don’t have to worry
about a phase shift, so
c = 0.
5. One period of the graph
is from 0 to , so we can
use that to calculate b.

19. Writing Equations From Graphs
 Example: Write an equation for the graph below.


2
period
b

 
2
b
2b
 1 sin2y x

20. Writing Equations From Graphs
 Example: Write an equation using cosine for the graph.

21. Writing Equations From Graphs
 Example: Write an equation using cosine for the graph.
1. d = ‒2,
2. This time we have a
phase shift. Since we
have to use cosine, we
will shift the graph over
/4 to the right.

1
2
a
4
c

 

22. Writing Equations From Graphs
 Example: Write an equation using cosine for the graph.
3. Since cosine normally
starts above the x-axis,
this graph has a
negative a.
4. The period goes from 0
to 2, so b is 1.
 
    
 
1
2 cos
2 4
y x

23. Writing Equations From Graphs
 Example: Write an equation using cosine for the graph.
Alternatively, we could
have kept a positive and
done a phase shift of 5/4.
1 5
2 cos
2 4
y x
 
    
 

24. Writing a Function From a Graph
 Example: Write a function to describe the graph below.

25. Writing a Function From a Graph
 Example: Write a function to describe the graph below.
c = 1
a = 1.5
Period goes from 0 to , so
The graph crosses the
midline at the y-axis, so we
will use sine, and d = 0.

 

2
2b
  1 1.5sin2f x x

26. Classwork
 College Algebra
 Page 606: 24-30, page 594: 42-46, page 582: 70-88
(all evens)