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.
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
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
16.
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…