The document describes properties of trigonometric and exponential functions. It discusses: 1) The graphs of sine and cosine functions have a domain of real numbers, range from -1 to 1, and cycle through these values every 2π units of x with a smooth curve. 2) Exponential functions with a base greater than 1 have a domain of all real numbers, range from 0 to infinity, and increase rapidly with horizontal asymptote at y=0. 3) The amplitude and period of trigonometric functions can be adjusted through multiplication and composition with other functions.

1. GRAPHS OF TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS

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6. The cycle repeats itself indefinitely in both directions of the
x-axis.
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
1. The domain is the set of real numbers.
5. Each function cycles through all the values of the range
over an x-interval of .

2
2. The range is the set of y values such that .
1
1 

 y

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Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.
0
-1
0
1
0
sin x
0
x
2

2
3

2

Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y
2
3


 2



2
2
3

2

2
5
1

1
x
y = sin x

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Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.
1
0
-1
0
1
cos x
0
x
2

2
3

2

Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y
2
3


 2



2
2
3

2

2
5
1

1
x
y = cos x

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y
1
1

2

3

2
x
 
3

2

 
4
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min
x-int
max
3
0
-3
0
3
y = 3 cos x
2

0
x 2

2
3
(0, 3)
2
3
( , 0)
( , 0)
2


2
( , 3)

( , –3)

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The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
2
3
2

4
y
x
4


2

y = –4 sin x
reflection of y = 4 sin x y = 4 sin x
y = sin x
2
1
y = sin x
y = 2 sin x

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2
y
2

6

x
2


6
5
3

3
2
6

6

3

2

3
2
0
2
0
–2
0
y = –2 sin 3x
0
x
Example: Sketch the graph of y = 2 sin(–3x).
Rewrite the function in the form y = a sin bx with b > 0
amplitude: |a| = |–2| = 2
Calculate the five key points.
(0, 0) ( , 0)
3

( , 2)
2

( ,-2)
6

( , 0)
3
2
Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x
period:
b

2 
2
3
=

8. Exponential Functions

9. EXPONENTIAL FUNCTIONS
In this chapter you will study two types of nonalgebraic functions—
exponential functions and logarithmic functions.

10. EXPONENTIAL FUNCTIONS
Note that in the definition of an exponential function, the base a = 1 is
excluded because it yields
f(x) = 1x = 1.
This is a constant function, not an exponential function.
Constant
function

11. The graph of f(x) = abx, b > 1
y
x
(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote
y = 0
4
4

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13. EXAMPLE 2 – GRAPHS OF Y = AX
In the same coordinate plane, sketch the graph of each function by
hand.
a. f(x) = 2x b. g(x) = 4x
Solution:
Figure 3.1

14. EXAMPLE 2 – SOLUTION
Note that both graphs are increasing. Moreover, the graph of g(x) =
4x is increasing more rapidly than the graph of
f(x) = 2x . You can tell if you compare the y values in the table below.
cont’d

15. Example: Sketch the graph of g(x) = 4x-3 + 3.
State the domain and range.
x
y
Make a table.
Domain: (–, )
Range: (3, ) or y > 3
2
–2
4
x y
3 4
2 3.25
1 3.0625
4 7
5 19

17. x y
-2
-1
0
1
2
Complete the table.
Substitute -2 for x
y = 9
Continue
substituting
numbers for x until
the table is
complete then graph
the points and draw
the graph
9
3
1
0.3
0.1

18. x y
-3
-2
-1
0
1
Complete the table
and sketch the
graph.
-1
5
-2.5
-2.9
-2.97
This graph has a
horizontal
asymptote at y = -3

19. Example 2
Graph each function.
x y
-1
0
1
2
3
4
Complete the table.
Substitute -1 for x
y = -0.004
-0.004
Continue
substituting
numbers for x until
the table is complete
then graph the points
and draw the graph
-0.02
-0.1
-0.5
-2.5
-12.5
This graph is reflected because the ½ is
negative – the graph does NOT cross or
touch the x-axis

20. The graph of f(x) = ex
y
x
2
–2
2
4
6
x f(x)
-2 0.14
-1 0.38
0 1
1 2.72
2 7.39
e  2.718281828…