Exponential Functions: Comparing Two Payment Options, Study notes of Pre-Calculus

Notes for a mathematics class on exponential functions, including an introduction to the concept and a comparison of two payment options. Students are asked to determine formulas for each option, calculate the amount of money received for attending a certain number of classes, and graph the money made over 30 classes. The document also reviews laws of exponents and properties of exponential functions.

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Pre 2010

Uploaded on 08/03/2009

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MATH 1113 PreCalculus Section 5.3 notes
Introduction to Exponential functions
Suppose that are dreaming and I offer to pay you each day for attending class! Being even more generous in
dreamland, I offer you a choice:
Choice 1 – Receive $1,000 per class attended.
Choice 2 – Receive 2 cents on the first day you attend class,
4 cents the second day, 8 cents the third day, and so on.
Before we go on, which choice do you choose? _____
1. Determine a formula for choice 1
2. What kind of equation did you create? ___________
3. How much money would you get for attending:
a) 10 classes ________________________
b) 20 classes ________________________
c) 30 classes ________________________
4. Draw a graph of the money you make over 30
classes.
y
x
1. Let’s determine a formula for the amount of money
you would receive on the xth day.
Class Amount received Formula?
0
1 2 cents
2 4 cents
3 8 cents
4
5
.
.
.
x
2. The equation _________________ is an
_________________ function.
3. How much money would you get for attending:
a) 10 classes _______________________
a) 20 classes ______________________
b) 30 classes _______________________
4. Draw a graph of the money you make over 30
classes.
y
x
pf3
pf4

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MATH 1113 PreCalculus Section 5.3 notes

Introduction to Exponential functions

Suppose that are dreaming and I offer to pay you each day for attending class! Being even more generous in

dreamland, I offer you a choice:

Choice 1 – Receive $1,000 per class attended. Choice 2 – Receive 2 cents on the first day you attend class,

4 cents the second day, 8 cents the third day, and so on.

Before we go on, which choice do you choose? _____

  1. Determine a formula for choice 1
  2. What kind of equation did you create? ___________
  3. How much money would you get for attending:

a) 10 classes ________________________

b) 20 classes ________________________

c) 30 classes ________________________

  1. Draw a graph of the money you make over 30

classes.

y

x

  1. Let’s determine a formula for the amount of money you would receive on the x th day.

Class Amount received Formula? 0 1 2 cents 2 4 cents 3 8 cents 4 5... x

  1. The equation _________________ is an _________________ function.
  2. How much money would you get for attending:

a) 10 classes _______________________

a) 20 classes ______________________

b) 30 classes _______________________

  1. Draw a graph of the money you make over 30 classes. y

x

Review from algebra class Laws of Exponents

If s, t, a, and b are real numbers, with a > 0 and b > 0, then

____

1 ___ ____

______ ( ) ______ ( ) ________

0  = 

a a

a

a a a ab

s s s

s t st s

1 2 4 _____________ 2. − =

13 ( 27 ) _______________ 3. =

3 6 1 3 ( x y ) ___________

For more review of these rules see the appendix A pages A85-A87.

Exponential Functions

The exponential function f with base a is ________________, where a > 0 and a ≠ 1.

Consider x f ( x )= 3 , x g ( x )= 7 ,

x h x  

( ) , and

x r x  

Observations about graphs of exponential functions of the form

x f ( x )= a

  1. All graphs pass through the point ______________, which is the __-intercept.
  2. ___________ is a horizontal asymptote.
  3. For bases > 1 or for a > 1, the functions are _______________ (increasing/decreasing)
  4. For bases between 0 and 1 or 0 < a < 1, the functions are __________________ (increasing/decreasing)
  5. For bases > 1, the ______________(larger/smaller) the base, the steeper the graph.
  6. For bases between 0 and 1, the ______________(larger/smaller) the base, the steeper the graph.
  7. All graphs have domain _____________.
  8. All graphs have range _______________.
  9. All graphs have ______ x -intercept(s).
  10. All graphs pass through the points _________, __________, and __________.

Graph

x

x

x

f x e

f x

f x

A Property of Exponential functions a^ a u v

u v If = ,then =

Use the above property to solve the following equations:

1

x

x x 4 2

2 = 3) 4 2

1

1

^ = 

  

x

5.3 Assignment 1-10ALL, 11, 17, 21, 23, 29-36ALL, 37-40ALL, 45-48ALL, 53-75ODD

3 - Solve: 2 32 x =

( )

2 1 4 3

1 Solve: x x e (^) x e e

− − = ⋅