Advanced Algebra with Trig
Advanced Functions NAME ____________________
4.2 DATE __________ PER _____
Exponential Functions and Their Applications
Definition of an Exponential Function with base b
is defined by [pic] where [pic], [pic], and x is a real number.
EX 1] Graph the exponential function [pic].
| x |[pic] | [pic] |
|[pic] | | |
|[pic] | | |
| 0 | | |
| 1 | | |
| 2 | | |
NOTE: Properties of the graph of [pic].
• The y-intercept is __________.
• The graph passes through __________.
• As x decreases without bound, _______________.
• As x increases without bound, _______________.
• The graph of [pic] is a _______________, _______________, _______________ curve.
EX 2] Graph the exponential function [pic].
| x |[pic] | [pic] |
|[pic] | | |
|[pic] | | |
| 0 | | |
| 1 | | |
| 2 | | |
NOTE: Properties of the graph of [pic].
• The y-intercept is __________.
• The graph passes through __________.
• As x decreases without bound, _______________.
• As x increases without bound, _______________.
• The graph of [pic] is a _______________, _______________, _______________ curve.
Summary of the Properties of [pic]
For positive real numbers b, [pic], the exponential function defined by [pic] has the following properties:
1) The function f is a one-to-one function.
Domain: __________________________ Range: __________________________
2) The graph of f is a smooth continuous curve with a y-intercept of __________ , and
the graph passes through __________.
3) If [pic], f is an increasing function and the graph of f is asymptotic to the negative x-axis.
This means As [pic]_____ , and as [pic]_____ .
4) If [pic], f is a decreasing function and the graph of f is asymptotic to the positive x-axis.
This means As [pic]_____ , and as [pic]_____ .
[pic] where [pic] [pic] where [pic]
EX 3] What is the x-intercept of the graph of [pic]?
EX 4] Explain how to use the graph of [pic] to produce the graph of [pic].
EX 5] Explain how to use the graph of [pic] to produce the graph of [pic].
The number e is defined as the number that [pic] approaches as n increases without bound.
[pic]
The Natural Exponential Function
the function defined by [pic] for all real numbers x
EX 6] Graph the exponential function [pic].
| x |[pic] | [pic] |
|[pic] | | |
|[pic] | | |
| 0 | | |
| 1 | | |
| 2 | | |
EX 7] Graph [pic] and [pic] on the graph above.
How do the graphs of [pic] and [pic] compare with the graph of [pic]?
EX 8] Graph [pic] and [pic] on the graph provided.
What happened?
Graph [pic] What happened?
What can you conclude?
Homework: Worksheet 4.2 HW Show all work to earn full credit!
Advanced Functions NAME ____________________
4.2 Homework DATE __________ PER _____
Exponential Functions and Their Applications
Show all work to earn full credit!
#1-4 Evaluate the exponential function for the given x-values.
1) [pic]; When [pic], f(x) = _________. When [pic], f(x) = _________.
2) [pic]; When [pic], f(x) = _________. When [pic], f(x) = _________.
3) [pic]; When [pic] , f(x) = _________. When [pic], f(x) = _________.
4) [pic]; When [pic], f(x) = _________. When [pic], f(x) = _________.
#5-7 Use a TI-83 to evaluate the exponential function for the given x-value.
Round the answer to the nearest hundredth.
5) [pic]; [pic]
6) [pic]; [pic]
7) [pic]; [pic]
#8 & 9 Sketch the graph of each function. Label key points as well as the function.
8) [pic] 9) [pic]
#10 & 11 Explain how to use the parent graph, f , to produce the second graph.
10) [pic]; [pic] 11) [pic]; [pic]
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