Chapter 1
Section 1.1
Scientific Notation
Powers of Ten
10 4  10000
10 3  1000
10 2  100
101  10
10 0  1
10 1  .1
10  2  .01
10 3  .001
10  4  .0001
Standard Scientific Notation
N x 10 n where 1 N  10 and n is an integers
Examples of numbers in scientific notation
3.4 x 10 8
4.17 x 1011
Using Scientific Notation
The population of Mexico City is about 23,000,000
To change the number into scientific notation you move the decimal place seven places to
get: 2.3 x 10 7
The speed the speed of light is 30,000,000,000 m/s. Write this number in scientific
notation.
Answer: 3.0 x 1010
Example 1
Convert .00000079 to scientific notation
Answer: 7.9 x 10 8
Example 2
Convert .000000000043 to scientific notation
Answer: 4.3 x 10 12
Example 3
Convert 5.1 x 10 8 to decimal notation.
Answer: 510,000,000
Example 4
Convert 3.11 x 10 5 to decimal notation.
Answer: 311,000
Using operations with scientific notation
Multiplication with scientific notation
Example 5
Simplify (6.1 x 106 )(7.8 x 107 )
(6.1 x 10 6 )(7.8 x 10 7 )
(6.1)(7.8) x 10 67
47.58 x 1013
4.758 x 1014
Example 6
Simplify (3 x 1010 )(7 x 10 7 )
(3 x 1010 )(7 x 10 7 )
(3)(7) x 10107
21 x 1017
2.1 x 1018
Example 7
Simplify (5 x 1010 )(4 x 10 6 )
(5 x 1010 )(4 x 10 6 )
(5)(4) x 10106
20 x 10 4
2.0 x 10 5
Division with scientific notation
Example 8
Simplify
4.2 x 1012
2.1 x 108
4.2 x 1012
2.1 x 10 8
4.2
x 10128
2.1
2 x 10 4
Example 9
Simplify
5.8 x 109
3.5 x 103
5.8 x 10 9
3.5 x 10 3
5.8
x 10 93
3.4
1.66 x 10 6
Example 10
Simplify
(4.2 x 1010 )(3.4 x 10 4 )
2 x 108
(4.2 x 1010 )(3.4 x 10 4 )
2 x 10 8
14.38 x 1014
2 x 10 8
7.19 x 10 6
Example 11
(2 x 105 )(1.2 x 108 )
Simplify
3 x 1010
(2 x 10 5 )(1.2 x 10 8 )
3 x 1010
2.4 x 1013
3 x 1010
.8 x 10 3
8.0 x 10 2
Example 12
The National debt is about 7.8 x 1012 , and there about 296,000,000 Americans. What
would be the debt per American citizen?
Convert the population of America to scientific notation. 296,000,000  2.96 x 10 8
Each American would owe:
$7.8 x 1012
 $2.635 x 10 4  $26,350
8
2.96 x 10
Estimation
Example 13
You make $13.85 per hour, about how much would make in a year assuming you work
40 hours a week?
Round $13.85 to $14.00
Salary per week: 40 x $14.00  $560
Salary per year: 52 x $560  $29,100
Example 14
A high graduate makes about an average of $25,000 per year while a college graduate
makes about an average of $40,000 per year.
High School Graduate : 40 x $25,000  $1,000,000
College Graduate : 40 x $40,000  $1,600,000
Section 1.2
Percents
Introduction to Basic Percents
The word percent translates to mean “out of one hundred”. A score of 85% on test means
that you scored 85 points out of 100 possible points on the test. If you scored 44 out of
50 points on a test, then this would be a percent value of 88%. This value can be
obtained by multiplying the numerator and denominator by 2 as shown in the next
illustration.
44 2(44) 88


 .88  88%
50 2(50) 100
Since a percent represents the amount out of a hundred, to change a percent to a decimal,
you simply drop the percent symbol and divide by 100 which can be done by moving the
decimal two placing to the left as shown in the next examples.
45
 .45
100
64.5
64.5% 
 .645
100
45% 
Basic Percent Problems
One of the basic uses of percents is to find the percent amount of a given number. For
example, how you would take 34% of 60? This would be done by changing 34% to .34,
and then multiplying by .34 by 60 as shown here:
34%  .34  (.34)(60)  20.4
Example 1
What is 46% of 90?
46%  .46
(.46)(90)  41.4
Mark up, mark down, and sales price
There are many common uses for percents in our society. As consumers, people use
percentages to find sales prices, mark up prices, and discount. In this section, we will
study how to use percents to compute discounts, mark up prices, sales prices, and sales
tax. The first of these topics we will explore are discount and sales price.
Discount
Discount = (Percent Mark Down)(Retail Price)
Sale Price
Sale Price = Retail Price – Discount
Example 2
A men’s sports jacket that has a retail price of $170 is discounted by 25%. What is the
sale’s price of the sports jacket?
Mark Down  25%  .25
Discount  .25$170  $42.50
Sales price  $170  $42.50  $127.50
Example 3
A pair of jeans that has a retail price of $55.00 is discounted at 30%. What is the sale’s
price of the jeans?
Mark Down  30%  .30
Discount  .30$55.00  $16.50
Sales price  $55  $16.50  $38.50
Example 4
The sale price of a VCR is $110.00. If the mark down is 30%, find the retail price of the
VCR.
Let x  original price
.30 x  discount
discount price  $110.00
x  .30 x  110.00
.70 x  110.00
.70 x 110.00

.70
.70
x  $157.14
Mark Up Price
When stores purchase items at a whole sale price, the retail price is computed by marking
up the whole sale cost using the given formulas.
Mark Up = (Percent Mark Up)(Whole Sale Price)
Retail Price = Whole Sale Price + Mark Up
Example 5
A store purchases DVD players at a whole sale price of $30 per unit which is to be
marked up by 80%. What will be the retail price of the DVD player?
percent mark up  80%  .80
mark up  (.80)($30.00)  $24.00
retail price  $30.00  $24.00  $54.00
Example 6
The whole sale price of a pair of jeans is $20.00. If the jeans are marked up by 65%,
what is the retail price of the jeans?
percent mark up  65%  .65
mark up  (.65)($20.00)  $13.00
retail price  $20.00  $13.00  $33.00
Example 7
The retail price of a new television that has been marked up by 75% is $300.00. Find the
whole sale price of the television.
Let x  original price
.75 x  discount
discount price  $300.00
x  .75 x  300.00
1.75 x  300.00
1.75 x 300.00

1.75
1.75
x  $171.43
Sales Tax
When items are purchased at a store or place of business, a state sale’s taxes is calculated
and added on the price of the item. The percent rate of sale’s tax in the United States is
determined by each state. For example the sales tax in Virginia is 4.5%. Some states
such as Delaware and Montana do not have any sale’s tax. The state sale’s tax is
calculated by multiplying the percent rate by the purchase price. The state sale’s tax is
then added on the purchase price of the item.
Sales Tax Formula
Sale’s Tax = (sale’s tax rate)(purchase price)
Example 8
The state sale’s tax rate in Virginia is 4.5%. Find the full cost to purchase a $50 pair of
shoes using the Virginia tax rate of 4.5%.
sales tax  (.045)(50)  $2.25
Cost including tax  $50.00  $2.25  $52.25
Example 9
The state sale’s tax rate in Ohio is 6%. Find the full cost to purchase the same pair of
shoes in problem 7 using the Ohio tax rate of 6%.
sales tax  (.06)(50)  $3.00
Cost including tax  $50.00  $3.00  $53.00
Problem Set (Section 1.2)
1)
a)
b)
c)
Find the discount on each item if the mark down rate is 5%.
$90.00
$25.00
$130.00
2) Find the discount on each item if the mark down rate is 15%.
a) $100.00
b) $45.00
c) $140.00
3) Find the sale’s price on each item given the mark down rate is 20%.
a) $120.00
b) $400.00
c) $215.00
4) Find the sale’s price on each item given the mark down rate is 15%.
a) $60.00
b) $130.00
c) $15.00
5) A pair of jeans that has a retail price of $42.00 is discounted at 25%. What is the
sale’s price of the jeans?
6) A women’s dress that has a retail price of $80 is discounted by 35%. What is the
sale’s price of the dress?
7) The sale price of a television is 200.00. If the mark down is 22%, find the retail
price of the television.
8) The sale price of a laptop computer is $1100.00. If the mark down is 10%, find the
retail price of the laptop computer.
9) Using a mark up rate of 30%, find the retail price given the whole sale price of
each item.
a) $140.00
b) $30.00
c) $75.00
10) Using a mark up rate of 45%, find the retail price given the whole sale price of
each item.
a) $200.00
b) $34.00
c) $124.00
11) The wholesale price of a pair of dress pants is $25.00. If the jeans are marked up
by 60%, what is the retail price of the pants?
12) The wholesale price of a CD player is $57.00. If the CD player is marked up by
30%, what is the retail price of the CD player?
13) The retail price of a pair of dress pants is $70.00. If the jeans are marked up by
25%, what is the whole sale price of the pants?
14) The retail price of a new television that has been mark up by 55% is $420.00.
Find the whole sale price of the television.
15) The sale’s tax rate in North Carolina is 4.5%. Find the total cost including sale’s
tax for purchasing each item.
a) $150.00
b) $340.00
16) The sale’s tax rate in Michigan is 6%. Find the total cost including sale’s tax for
purchasing each item.
a) $250.00
b) $420.00
Section 1.3
Introduction to Mathematical Modeling
Types of Modeling
1) Linear Modeling
2) Quadratic Modeling
3) Exponential Modeling
4) Logarithmic Modeling
Each type of modeling in mathematics is determined by the graph of equation for each
model. In the next examples, there is a sample graph of each type of modeling
Linear models are described by the following general graph
Quadratic models are described by the following general graph
Exponential models are described by the following general graph
Logarithmic Models are described by the following general graph.
Section 1.4
Linear Models
Before you can study linear models, you must understand so basic concepts in Algebra.
One of the main algebra concepts used in linear models is the slope-intercept equation of
a line. The slope intercept equation is usually expressed as follows:
Standard linear model
y  mx  b
m  slope
b  y  Intercept
In this equation the variable m represents the slope of the equation and the variable b
represents the y-intercept of the line. When studying linear models, you must understand
the concept of slope. Slope is usually defined as “rise over run” or “change in y over
change in x”. In general slope measures the rate in change. Thus, the idea of slope has
many applications in mathematics including velocity, temperature change, pay rates, cost
rates, and several other rates of change.
Slope
rise change in y

run change in x
y  y1
m 2
x 2  x1
Slope 
Basic Algebra Skills (Slope and y-intercept)
In next examples, we will find the slope of a line given two points on the line.
Example 1
Find the slope between the points (1,3) and (3,2)
m
y 2  y1 2  3  1
1



2
2
x 2  x1 3  1
Example 2
Find the slope between the points (2,3) and (4,6)
m
y 2  y1 6  3 3


x 2  x1 4  2 2
Slope and y-intercept also can be found from the equation in slope-intercept, as shown in
this next example. Notice that the equation is written in slope-intercept form.
Example 3
Find the slope and y-intercept
y  3x  2
m3
b  2
If the equation is not written in slope intercept form, it can be rearranged to slopeintercept form by solving the equation for y. This procedure is shown in the next two
examples.
Example 4
Find the slope and y-intercept
2x  3y  6
2 x  2 x  3 y  2 x  6
3 y  2 x  6
3y  2x 6


3
3
3
2
y  x2
3
2
3
b  2
m
Example 5
Find the slope and y-intercept
3 x  5 y  10
3 x  3 x  5 y  3 x  10
 5 y  3 x  10
 5 y  3 x 10


5
5 5
3
y  x2
5
3
5
b  2
m
Example 6
3
x2
2
First construct a table using 4 arbitrary values of x, and then substitute these x values to
3
the equation y  x  2 to get the corresponding y values.
2
x
3
y  x2
2
1
3
3
1
y  (1)  2   2  
2
2
2
2
3
y  (2)  2  3  2  1
2
3
3
9
5
y  (3)  2   2 
2
2
2
4
3
y  (4)  2  6  2  4
2
Graph the equation y 
Next make point using the four points in the above table.
4
2
-5
5
-2
-4
-6
Applications of Linear Equations
Example 6 (Temperature conversion)
9
F  C  32
5
9
a) Sketch a graph of F  C  32
5
C
10
20
30
40
9
F  C  32
5
9
F  (10)  32  9(2)  32  18  32  50
5
9
F  (20)  32  9(4)  32  36  32  68
5
9
F  (30)  32  9(6)  32  54  32  86
5
9
F  (40)  32  9(8)  32  72  32  104
5
b) Use the model to convert 120 degrees Celsius to degrees Fahrenheit.
9
F  C  32
5
9
F  (120)  32
5
F  216  32
F  248
c) Use the model to convert 212 degrees Fahrenheit to Celsius.
9
F  C  32
5
9
212  C  32
5
9
212  32  C  32  32
5
9
180  C
5
5
5 9
(180)   C
9
9 5
C  100 0 C
Example 7 (Business Applications)
The revenue of a company that makes backpacks is given by the formula R  21.50 x
where x represents the number of backpacks sold.
a) Graph the linear model R  21.50 x
X
10
20
30
40
R  21.50 x
R  21.50(10)  215
R  21.50(20)  430
R  21.50(30)  645
R  21.50(40)  860
b) Use the model to calculate the revenue for selling 50 backpacks
x  50
R  21.50 x  21.5(50)  $1075.0
c) What is the slope
m  $21.50
d) What is the meaning of the slope?
Cost per unit sold
Revenue made per backpack solid
Example 8 (Sales)
A salesperson is paid $100 plus $60 per sale each week. The model S  60 x  100 is
used to calculate the salesperson’s weekly salary where x is the number of sales per
week.
a) Graph S  60 x  100
x
2
4
6
8
S
S
S
S
S
 60 x  100
 60(2)  100  120  100  220
 60(4)  100  240  100  340
 60(6)  100  360  100  460
 60(8)  100  480  100  580
b) Use the model to calculate the salespersons weekly salary if he/she makes 8 sales.
S  60(8)  100  480  100  $580.00
c) What is the slope of the equation
m  60
$
sale
d) What is the meaning of the slope
Dollars per each sale
Example 9
Given the following data sketch a graph
Time
1 min
2 min
3 min
4 min
Temperature
30 C
70 C
110 C
14 0 C
Sketch a graph of the given data and then compute the slope of the resulting line.
12
10
8
(2,7)
6
4
(1,3)
2
-5
5
10
-2
Use the points (1,3) and (2,7) in the above graph to compute the slope
m
73 4
 4
2 1 1
15
Example 10
An approximate linear model that gives the remaining distance, in miles, a plane must
travel from Los Angeles to Paris given by d  6000  550t where d is the remaining
distance and t is the hours after the flight begins. Find the remaining distance to Paris
after 3 hours and 5 hours.
d  6000  550(3)
d  6000  1650
d  4350 miles
d  6000  550(5)
d  6000  2750
d  3250 miles
How long should it take for the plane to flight from Los Angeles to Paris?
0  6000  550t
0  550t  6000  550t  550t
550t  6000
550t 6000

550
550
t  10.9 hours
Problem Set 1.4
1) Find the slope between the points (1,1) and (3,5)
2) Find the slope between the points (0,0) and (4,5)
Given the equation, find the slope and y-intercept.
3
x2
4
4) 3x  4 y  6
5) 2 x  3 y  6
3) y 
Graph the following equations
6) y  3 x
7) y  x  5
1
8) y  x  1
4
9) y  6 x
Linear Models
10) The revenue of a company that makes backpacks is given by the formula R  34.50 x
where x represents the number of backpacks sold.
a) Graph the linear model R  34.50 x
b) Use the model to calculate the revenue for selling 40 backpacks?
c) What is the slope of the model?
d) What is the meaning of the slope?
11) A salesperson is paid $100 plus $30 per sale each week. The model S  30 x  100 is
used to calculate the salesperson’s weekly salary where x is the number of sales per
week.
a) Graph S  30 x  100
b) Use the model to calculate the salespersons weekly salary if he/she makes 8 sales.
c) What is the slope of the equation?
d) What is the meaning of the slope?
12) A salesperson is paid $200 plus $50 per sale each week. The model S  50 x  200 is
used to calculate the salesperson’s weekly salary where x is the number of sales per
week.
a) Graph S  50 x  200
b) Use the model to calculate the salespersons weekly salary if he/she makes 8 sales.
c) What is the slope of the equation?
d) What is the meaning of the slope?
13) An approximate linear model that gives the remaining distance, in miles, a plane must
travel from San Francisco to London given by d (t )  5500  500t where d (t ) is the
remaining distance and t is the hours after the flight begins. Find the remaining distance
to London after 2 hours and 4 hours.
Section 1.5
Quadratic Models
Graph of Quadratic Models
The graph of a quadratic model always results in a parabola. The general form of a
quadratic function is given in the following definition.
A quadratic function is a function where the graph is a parabola and the equation is of
the form: y  ax 2  bx  c where a  0
b
The x-coordinate of vertex is given by the equation: x  
2a
The vertex is the turning point on the graph of a parabola. If the parabola opens upward,
then the vertex is the lowest point of the graph. If the parabola opens downward, then the
vertex is the highest point on the graph. The direction of the parabola opens can be
determined by the sign of the “ x 2 ” term or the a term in the above equation. If a  0 ,
then the parabola open downward. Similarly if a  0 , then the parabola opens upward.
(See graphs below in figure 1-1)
Figure 1-1
A parabola where a  0 and the vertex is the lowest point on the graph
A parabola where a  0 and the vertex is the highest point on the graph
Here are some examples of finding the vertex and x-intercepts of an exponential
equation. The graph of the quadratic equation is also provided in these examples
Example 1
Find the vertex and x-intercepts of the quadratic equation, and then make a sketch of the
parabola.
y  x2  3
a  1, c  3
x
0
0
 0
2(1)
2
x-intercepts:
x2  3  0
x2  3
x2  3
x 3
( 3 ,0) and ( 3 ,0)
Graph for Example 1
Example 2
Find the vertex and x-intercepts of the quadratic equation, and then make a sketch of the
parabola.
y  x 2  3x
Vertex
x
3 3

2(1) 2
2
9
3
3 9 9
y     3     
4
2
2 4 2
x-intercepts
x 2  3x  0
x( x  3)  0
x  0 or x  3  0
x  0 x 3  0
x3
(0,0) and (3,0)
Graph of the function
Example 3
Find the vertex and x-intercepts of the quadratic equation, and then make a sketch of the
parabola.
y  3x 2  6 x
Vertex
x
 (6) 6
 1
2(3)
6
y  31  61  3  6  3
2
(1,3)
x-intercepts
3x 2  6 x  0
3x( x  2)  0
3x  0 or x  2  0
x 0 x20
x2
(0,0) and (2,0)
Graph of 3x 2  6 x  0
More about Quadratic Equations
In some instances, the quadratic equation will not factor properly. In this case, you must
use what is called the quadratic formula. In the next few examples, the quadratic formula
will be used to find the solutions of a quadratic equation.
The Quadratic Formula
The solution to the equation y  ax 2  bx  c is given by
x
 b  b 2  4ac
2a
Example 4
Solve x 2  5 x  7  0
a 1
b5
c  7
2
 b  b 2  4ac  5  5  4(1)(7)  5  25  28  5  53



x
2a
2(1)
2
2
Example 5
Solve x 2  7 x  9  0
x
 7  7 2  4(1)(9)  7  49  36  7  85


2( 7 )
2( 7 )
14
Example 6
At a local frog jumping contest. Rivet’s jump can be approximated by the equation
1
1
y   x 2  2 x and Croak’s jump can be approximate by y   x 2  4 x , where x = the
6
2
length of jump in feet and y = the height of the jump in feet.
a) Which frog can jump higher
1
2
2
Rivet’s vertex: x  

 6 Height: y   (6) 2  2(6)  6  12  6 ft
1
6
 1

2  
3
 6
1
4
4
Croak’s vertex: x  

 4 Height: y   (4) 2  4(4)  8  16  8 ft
2
 1  1
2  
 2
Croak can jump higher at 8 feet
b) Which frog can jump farther
Rivet’s can jump farther at 2(6 ft) = 12 feet
Graph of the frogs jumps
8
g x =
 
-1
2
x2+4 x
6
4
f x =
 
-1
6
x2+2x
2
-5
5
-2
Using the parabola to find the maximum or minimum value of a quadratic function
The parabola can be used to find either the maximum value or the minimum value of a
quadratic function. (See figure 1-1) This can simply be done by find the vertex of the
parabola. Remember as stated earlier the vertex will turn out to be either the highest
point on the curve or the lowest point on the curve. In the next examples, the vertex of
the parabola will be use to find the maximum value.
Example 7
The path of a ball thrown by a boy is given by the equation y  .04 x 2  1.5 x where x is
the horizontal distance the ball travels and y is the height of the ball. Find the maximum
height of the ball in yards.
Find the vertex of the ball
x
1.5
1.5

 18.75
2(.04) .08
y  .0418.75  1.5(18.75)  14.1  28.1  14 yards
2
Example 8
The path of a cannon ball is given by the equation y  .1x 2  6.0 x where x is the
horizontal distance the ball travels and y is the height of the cannon ball. Find the
maximum height of the cannon ball in feet.
Find the vertex of the cannon ball.
x
 6.0  6.0
2

 30  y  .130  6(30)  90  180  90 feet
2(.1)
 .2
Problem Set 15
Find the vertex and x-intercepts of the given parabola, and then make a sketch of
the parabola.
1)
2)
3)
4)
5)
6)
y  2x 2  4x
y  x2  4
y  x 2  2x  1
y  x 2  4x  3
y  x 2  16
y  3x 2  6 x
Quadratic Models
7) The path of a ball thrown by a baseball player is given by the equation
y  .02 x 2  1.6 x where x is the horizontal distance the ball travels and y is the height of
the ball. Find the maximum height of the ball in yards.
8) The path of a ball thrown by a boy is given by the equation y  .06 x 2  1.8 x where x
is the horizontal distance the ball travels and y is the height of the ball. Find the
maximum height of the ball in yards.
9) The path of a cannon ball is given by the equation y  .05 x 2  6.0 x where x is the
horizontal distance the ball travels and y is the height of the cannon ball. Find the
maximum height of the cannon ball in feet.
10) The path of a cannon ball is given by the equation y  .1x 2  8.0 x where x is the
horizontal distance the ball travels and y is the height of the cannon ball. Find the
maximum height of the cannon ball in feet.
Section 1.6
Exponential models
The exponential function
e  2.718 “The Euler number”
Example 1: Simplify the following exponential functions
1) e 2  7.39
2) e  3 
1
 .05
e3
1
3
3) e  1.40
The graph of the exponential function
Example 2 Graph y  e x
x
y  ex
-2
y  e 2  .14
-1
y  e 1  .37
0
y  eo  1
1
y  e1  2.7
2
y  e 2  7.4
Example 3
Graph y  10e .2 x
x
-2
Y
y  10e .2 ( 2)  10e .4  6.7
-1
y  10e .2 ( 1)  10e .2  8.2
0
y  10e .2( 0)  10e 0  10
1
y  10e .2 (1)  10e .2  12.2
2
y  10e .2 ( 2 )  10e .4  14.9
Exponential Models
Exponential models are used to predict human populations, animal populations, money
growth, pollution growth, and other aspects of society that fit exponential models. The
variable of an exponential model is found in the exponent of the equation.
Exponential Growth
P  P0 (1  r ) t
P  New Value
P0  Original Value
r  rate
t  time
Example 4
The population of the United States is 290 million, what would be the population of the
U. S. be in 20 years if its population would growth at a steady rate of .7 % for 20 years?
P  P0 (1  r ) t
P0  290,000,000
r  .7%  .007
t  20
P  290000000(1  .007) 20  290000000(1.007) 20  333416746
Example 5
The population of Blacksburg, Virginia is 41,000, what would be the population in 10
years if Blacksburg would grow at a rate of 1.1 % per year?
P  P0 (1  r )t
P0  41000
r  1.1%  .011
t  10
P  41000(1  .011)10  41000(1.011)10  45740
Example 6
In 1995 the United States had greenhouse emissions of about 1400 million tons, where as
China had greenhouse emissions of about 850 million tons. If in the next 25 years China
greenhouse emission grew by 4 percent and the U. S. greenhouse emission grew by 1.3
percent, what would the emissions in tons for both countries in 2020?
U . S . Emissions in 2020
P  P0 (1  r )t
P0  1400 million
r  1.3%  .013
t  25
P  1400(1  .013) 25  1400(1.013) 25  1933 million tons
China ' s Emissions in 2020
P  P0 (1  r )t
P0  850 million
r  4.0%  .04
t  25
P  850(1  .04) 25  850(1.04) 25  2265 million tons
Example 7
Using the exponential growth formula, find the amount of money that you would have in
a bank account if you deposited $3,000 in the account for 15 years at 1.1 % interest rate?
P  P0 (1  r ) t
P0  3000
r  1.1%  .011
t  15
P  3000(1  .011)15  3000(1.011)15  $3482.91
Exponential decay
Exponential decay models are use to measure radioactive decay, decreasing populations,
Half-life, and other elements that fit an exponential model. Again, the one variable in an
exponential decay models in found in the exponent.
Exponential Decay Formula
P  P0 (1  r ) t
P  New Value
P0  Original Value
r  rate
t  time
Example 8
A certain population of black bears in the eastern United States has been decreasing by
3.1 percent per year. If this trend keeps up, what will be the population of bears in 20
years if there are currently 1000 bears.
P  P0 (1  r )t
P0  1000
r  3.1%  .031
t  20
P  1000(1  .031) 20  1000(.969) 20  533
Example 9
A certain isotope decreases at a rate of 5% per year. It there is currently 340 grams of the
isotope, how many grams of the isotope will there be in 20 years?
P  P0 (1  r )t
P0  340
r  5%  .05
t  25
P  340(1  .05) 20  340(.95) 20  122 grams
Problem Set 1.6
Exponential Functions
Evaluate using a calculator
1) e 2
1
2) e
2
3) 2e
4
3
Graph the following functions
4)
5)
6)
7)
y  3x
y  ex 1
y  2e x
y  e2x
Growth Models (Show Work)
8) The current population of Germany is 80,000,000. What would be the population of
Germany in 10 years if its population would growth at a steady rate of .9 % for 10 years?
9) The current population of Salem, Virginia is 25,000. What would be the population of
Salem in 5 years if Salem would grow at a rate of 1.2 % per year?
10) Using the exponential growth formula, find the amount of money that you would
have in a bank account if you deposited $10,000 in the account for 10 years at 1.6 %
interest rate?
11) A certain rabbit population is modeled by the equation P  2000e .03t where t is the
time in months. Use the model to predict the population after 20 months.
Decay Models
11) A certain population of Panda Bears in China has been decreasing by 1.0 percent per
year. If this trend keeps up, what will be the population of Panda Bears in 10 years if
there are currently 2000 bears?
12) A certain isotope decreases at a rate of 4% per year. It there is currently 220 grams of
the isotope, how many grams of the isotope will there be in 25 years?
Section 1.7
Basic Logarithms
Definition of a Logarithm
log b a  x  b x  a
Example 1
i)
Write 35  243 as a logarithmic expression.
35  243  log 3 243  5
ii)
Write 5 4  625 as a logarithmic expression.
5 4  625  log 5 625  4
Example 2
i)
Write log 4 16  2 as exponential expression.
log 4 16  2  4 2  16
ii)
Write log10 10,000  4 as an exponential expression.
log10 10,000  4  10 4  10,000
Log base ten
Another way of writing log10 1000 is log1000 .
The way we find the answer to log1000 is to ask the question of 10 raised to what power
gives you 1000? Since we know that 10 4  1000 , the answer is 4.
Example 3
i)
Find log 100,000
Since 10 5  100,000 , log 100,000  5
ii)
Find log 100
Since 10 2  100 , log 100  2
Example 4
Use a scientific calculator to evaluate the following logarithms
i) log 567
Answer: log 567 = 2.754
ii) log 30890
Answer: log 30890 = 4.490
iii) log 456782
Answer: log 456782 = 5.660
Graph of basic logarithms
________________________________________________________________________
Example 5
Graph y  log 6 x
X
2
10
20
40
Y
y  log(6(2))  log(12)  1.07
y  log(6(10))  log(60)  1.8
y  log(6(20))  log(120)  2.1
y  log(6(40))  log(240)  2.4
Plot the given values from the table gives the following graph
Example 6
Graph y  5 log( x  1)
X
2
10
20
40
y
y  5 log(2  1)  5 log(3)  2.4
y  5 log(10  1)  5 log(11)  5.2
y  5 log(20  1)  5 log(21)  6.6
y  5 log(40  1)  5 log(41)  8.1
Plot the given values from the table gives the following graph
Example 7 (Using logarithmic models to model height)
A logarithmic model to approximate the percentage P of an adult height a male has
reached at an age A form 13 and 18 is P  16 log( A  12)  84
1) Sketch a graph of this function.
P
13
14
15
18
A
P  16 log(13  12)  84  84
P  16 log(14  12)  84  16 log(2)  84  4.8  84  88.8
P  16 log(15  12)  82  16 log(3)  84  7.6  84  90.6
P  16 log(18  12)  84  16 log(6)  84  12.5  84  96.5
Plot the given values from the table gives the following graph
2) What does the graph tell you about the height of male after age of 18?
Usually males stop growing after age 18
3) Use the model to compute the average height of a 16 year old male.
P  16 log(16  12)  84  16. log(4)  84  9.6  84  93.6
93.6%
Example 8
Use the following model for $1000 invested in saving account given by
n  694.2  231.4 log( A) , to find the amount of time (n) for the amount of money A to
grow to $100,000.
n  694.2  231.4 log 100000
n  694.2  231.4(5)
n  694.2  1157
n  462.8
Problem Set 1.7
I)
Write as a logarithmic expression.
1) e 2  a
2) 2 6  64
II)
Write as an exponential expression.
1) log 3 81  4
2) ln x  y
III)
Graph each logarithmic equation.
1) y  ln x
2) y  ln x  5
3) y  ln2 x  6 