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## The Cartesian Coordinate System and Straight lines

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Title: The Cartesian Coordinate System and Straight lines

1
2
Functions and Their Graphs
• The Cartesian Coordinate System and
Straight lines
• Equations of Lines
• Functions and Their Graphs
• The Algebra of Functions
• Linear Functions
• Functions and Mathematical Models

2
2.1
• The Cartesian Coordinate System and Straight lines

3
The Cartesian Coordinate System
• We can represent real numbers geometrically by
points on a real number, or coordinate, line

4
The Cartesian Coordinate System
• The Cartesian coordinate system extends this
concept to a plane (two dimensional space) by

4 3 2 1 1 2 3 4
5
The Cartesian Coordinate System
• The horizontal line is called the x-axis, and the
vertical line is called the y-axis.

y
4 3 2 1 1 2 3 4
x
6
The Cartesian Coordinate System
• The point where these two lines intersect is
called the origin.

y
4 3 2 1 1 2 3 4
Origin
x
7
The Cartesian Coordinate System
• In the x-axis, positive numbers are to the right
and negative numbers are to the left of the
origin.

y
4 3 2 1 1 2 3 4
Positive Direction
Negative Direction
x
8
The Cartesian Coordinate System
• In the y-axis, positive numbers are above and
negative numbers are below the origin.

y
4 3 2 1 1 2 3 4
Positive Direction
x
Negative Direction
9
The Cartesian Coordinate System
• A point in the plane can now be represented
uniquely in this coordinate system by an ordered
pair of numbers (x, y).

y
( 2, 4)
4 3 2 1 1 2 3 4
(4, 3)
x
(3, 1)
(1, 2)
10
The Cartesian Coordinate System
• The axes divide the plane into four quadrants as
shown below.

y
4 3 2 1 1 2 3 4
x
11
Slope of a Vertical Line
• Let L denote the unique straight line that passes
through the two distinct points (x1, y1) and (x2,
y2).
• If x1 x2, then L is a vertical line, and the
slope is undefined.

y
L
(x1, y1)
(x2, y2)
x
12
Slope of a Nonvertical Line
• If (x1, y1) and (x2, y2) are two distinct points
on a nonvertical line L, then the slope m of L is
given by

y
L
(x2, y2)
y2 y1 ?y
(x1, y1)
x2 x1 ?x
x
13
Slope of a Nonvertical Line
• If m gt 0, the line slants upward from left to
right.

y
L
m 1
?y 1
?x 1
x
14
Slope of a Nonvertical Line
• If m gt 0, the line slants upward from left to
right.

y
L
m 2
?y 2
?x 1
x
15
Slope of a Nonvertical Line
• If m lt 0, the line slants downward from left to
right.

y
m 1
?x 1
?y 1
x
L
16
Slope of a Nonvertical Line
• If m lt 0, the line slants downward from left to
right.

y
m 2
?x 1
?y 2
x
L
17
Examples
• Sketch the straight line that passes through the
point (2, 5) and has slope 4/3.
• Solution
• Plot the point (2, 5).
• A slope of 4/3 means that if x increases by 3,
y decreases by 4.
• Plot the resulting point (5, 1).
• Draw a line through the two points.

y
6 5 4 3 2 1
?x 3
(2, 5)
?y 4
(5, 1)
x
1 2 3 4 5 6
L
18
Examples
• Find the slope m of the line that goes through
the points (1, 1) and (5, 3).
• Solution
• Choose (x1, y1) to be (1, 1) and (x2, y2) to be
(5, 3).
• With x1 1, y1 1, x2 5, y2 3, we find

19
Examples
• Find the slope m of the line that goes through
the points (2, 5) and (3, 5).
• Solution
• Choose (x1, y1) to be (2, 5) and (x2, y2) to be
(3, 5).
• With x1 2, y1 5, x2 3, y2 5, we find

20
Examples
• Find the slope m of the line that goes through
the points (2, 5) and (3, 5).
• Solution
• The slope of a horizontal line is zero

y
6 4 3 2 1
(3, 5)
(2, 5)
L
m 0
x
2 1 1 2 3 4
21
Parallel Lines
• Two distinct lines are parallel if and only if
their slopes are equal or their slopes are
undefined.

22
Example
• Let L1 be a line that passes through the points
(2, 9) and (1, 3), and let L2 be the line that
passes through the points ( 4, 10) and (3, 4).
• Determine whether L1 and L2 are parallel.
• Solution
• The slope m1 of L1 is given by
• The slope m2 of L2 is given by
• Since m1 m2, the lines L1 and L2 are in fact
parallel.

23
2.2
• Equations of Lines

24
Equations of Lines
• Let L be a straight line parallel to the y-axis.
• Then L crosses the x-axis at some point (a, 0) ,
with the x-coordinate given by x a, where a is
a real number.
• Any other point on L has the form (a, ), where
is an appropriate number.
• The vertical line L can therefore be described as
• x a

y
L
(a, )
(a, 0)
x
25
Equations of Lines
• Let L be a nonvertical line with a slope m.
• Let (x1, y1) be a fixed point lying on L, and let
(x, y) be a variable point on L distinct from
(x1, y1).
• Using the slope formula by letting (x, y) (x2,
y2), we get
• Multiplying both sides by x x1 we get

26
Point-Slope Form
• An equation of the line that has slope m and
passes through point (x1, y1) is given by

27
Examples
• Find an equation of the line that passes through
the point (1, 3) and has slope 2.
• Solution
• Use the point-slope form
• Substituting for point (1, 3) and slope m 2, we
obtain
• Simplifying we get

28
Examples
• Find an equation of the line that passes through
the points (3, 2) and (4, 1).
• Solution
• The slope is given by
• Substituting in the point-slope form for point
(4, 1) and slope m 3/7, we obtain

29
Perpendicular Lines
• If L1 and L2 are two distinct nonvertical lines
that have slopes m1 and m2, respectively, then L1
is perpendicular to L2 (written L1 - L2) if and
only if

30
Example
• Find the equation of the line L1 that passes
through the point (3, 1) and is perpendicular to
the line L2 described by
• Solution
• L2 is described in point-slope form, so its slope
is m2 2.
• Since the lines are perpendicular, the slope of
L1 must be
• m1 1/2
• Using the point-slope form of the equation for L1
we obtain

31
Crossing the Axis
• A straight line L that is neither horizontal nor
vertical cuts the x-axis and the y-axis at, say,
points (a, 0) and (0, b), respectively.
• The numbers a and b are called the x-intercept
and y-intercept, respectively, of L.

y
y-intercept
(0, b)
x-intercept
x
(a, 0)
L
32
Slope-Intercept Form
• An equation of the line that has slope m and
intersects the y-axis at the point (0, b) is
given by
• y mx b

33
Examples
• Find the equation of the line that has slope 3
and y-intercept of 4.
• Solution
• We substitute m 3 and b 4 into y mx b
and get
• y 3x 4

34
Examples
• Determine the slope and y-intercept of the line
whose equation is 3x 4y 8.
• Solution
• Rewrite the given equation in the slope-intercept
form.
• Comparing to y mx b, we find that m ¾ and
b 2.
• So, the slope is ¾ and the y-intercept is 2.

35
Applied Example
• Suppose an art object purchased for 50,000 is
expected to appreciate in value at a constant
rate of 5000 per year for the next 5 years.
• Write an equation predicting the value of the art
object for any given year.
• What will be its value 3 years after the
purchase?
• Solution
• Let x time (in years) since the object was
purchased
• y value of object (in dollars)
• Then, y 50,000 when x 0, so the y-intercept
is b 50,000.
• Every year the value rises by 5000, so the slope
is m 5000.
• Thus, the equation must be y 5000x 50,000.
• After 3 years the value of the object will be
65,000
• y 5000(3) 50,000 65,000

36
General Form of a Linear Equation
• The equation
• Ax By C 0
• where A, B, and C are constants and A and B are
not both zero, is called the general form of a
linear equation in the variables x and y.

37
General Form of a Linear Equation
• An equation of a straight line is a linear
equation conversely, every linear equation
represents a straight line.

38
Example
• Sketch the straight line represented by the
equation
• 3x 4y 12 0
• Solution
• Since every straight line is uniquely determined
by two distinct points, we need find only two
such points through which the line passes in
order to sketch it.
• For convenience, lets compute the x- and
y-intercepts
• Setting y 0, we find x 4 so the x-intercept
is 4.
• Setting x 0, we find y 3 so the y-intercept
is 3.
• Thus, the line goes through the points (4, 0) and
(0, 3).

39
Example
• Sketch the straight line represented by the
equation
• 3x 4y 12 0
• Solution
• Graph the line going through the points (4, 0)
and (0, 3).

y
L
1 1 2 3 4
(4, 0)
x
1 2 3 4 5 6
(0, 3)
40
Equations of Straight Lines
• Vertical line x a
• Horizontal line y b
• Point-slope form y y1 m(x x1)
• Slope-intercept form y mx b
• General Form Ax By C 0

41
2.3
• Functions and Their Graphs

42
Functions
• A function f is a rule that assigns to each
element in a set A one and only one element in a
set B.
• The set A is called the domain of the function.
• It is customary to denote a function by a letter
of the alphabet, such as the letter f.
• If x is an element in the domain of a function f,
then the element in B that f associates with x is
written f(x) (read f of x) and is called the
value of f at x.
• The set B comprising all the values assumed by y
f(x) as x takes on all possible values in its
domain is called the range of the function f.

43
Example
• Let the function f be defined by the rule
• Find f(1)
• Solution

44
Example
• Let the function f be defined by the rule
• Find f( 2)
• Solution

45
Example
• Let the function f be defined by the rule
• Find f(a)
• Solution

46
Example
• Let the function f be defined by the rule
• Find f(a h)
• Solution

47
Applied Example
• ThermoMaster manufactures an indoor-outdoor
thermometer at its Mexican subsidiary.
• Management estimates that the profit (in dollars)
realizable by ThermoMaster in the manufacture and
sale of x thermometers per week is
• Find ThermoMasters weekly profit if its level of
production is
• 1000 thermometers per week.
• 2000 thermometers per week.

48
Applied Example
• Solution
• We have
• The weekly profit by producing 1000 thermometers
is
• or 2,000.
• The weekly profit by producing 2000 thermometers
is
• or 7,000.

49
Determining the Domain of a Function
• Suppose we are given the function y f(x).
• Then, the variable x is called the independent
variable.
• The variable y, whose value depends on x, is
called the dependent variable.
• To determine the domain of a function, we need to
find what restrictions, if any, are to be placed
on the independent variable x.
• In many practical problems, the domain of a
function is dictated by the nature of the problem.

50
Applied Example Packaging
• An open box is to be made from a rectangular
piece of cardboard 16 inches wide by cutting away
identical squares (x inches by x inches) from
each corner and folding up the resulting flaps.

x
10 10 2x
x
16 2x
x
x
16
51
Applied Example Packaging
• An open box is to be made from a rectangular
piece of cardboard 16 inches wide by cutting away
identical squares (x inches by x inches) from
each corner and folding up the resulting flaps.
• Find the expression that gives the volume V of
the box as a function of x.
• What is the domain of the function?
• The dimensions of the resulting box are

x
10 2x
16 2x
52
Applied Example Packaging
• Solution
• a. The volume of the box is given by multiplying
its dimensions (length ? width ? height), so

x
10 2x
16 2x
53
Applied Example Packaging
• Solution
• b. Since the length of each side of the box must
be greater than or equal to zero, we see that
• must be satisfied simultaneously. Simplified
• All three are satisfied simultaneously provided
that
• Thus, the domain of the function f is the
interval 0, 5.

54
More Examples
• Find the domain of the function
• Solution
• Since the square root of a negative number is
undefined, it is necessary that x 1 ? 0.
• Thus the domain of the function is 1,?).

55
More Examples
• Find the domain of the function
• Solution
• Our only constraint is that you cannot divide by
zero, so
• Which means that
• Or more specifically x ? 2 and x ? 2.
• Thus the domain of f consists of the intervals (
?, 2), (2, 2), (2, ?).

56
More Examples
• Find the domain of the function
• Solution
• Here, any real number satisfies the equation, so
the domain of f is the set of all real numbers.

57
Graphs of Functions
• If f is a function with domain A, then
corresponding to each real number x in A there is
precisely one real number f(x).
• Thus, a function f with domain A can also be
defined as the set of all ordered pairs (x, f(x))
where x belongs to A.
• The graph of a function f is the set of all
points (x, y) in the xy-plane such that x is in
the domain of f and y f(x).

58
Example
• The graph of a function f is shown below

y
y
(x, y)
Range
x
x
Domain
59
Example
• The graph of a function f is shown below
• What is the value of f(2)?

y
4 3 2 1 1 2
x
1 2 3 4 5 6 7 8
(2, 2)
60
Example
• The graph of a function f is shown below
• What is the value of f(5)?

y
4 3 2 1 1 2
(5, 3)
x
1 2 3 4 5 6 7 8
61
Example
• The graph of a function f is shown below
• What is the domain of f(x)?

y
4 3 2 1 1 2
x
1 2 3 4 5 6 7 8
Domain 1,8
62
Example
• The graph of a function f is shown below
• What is the range of f(x)?

y
4 3 2 1 1 2
Range 2,4
x
1 2 3 4 5 6 7 8
63
Example Sketching a Graph
• Sketch the graph of the function defined by the
equation
• y x2 1
• Solution
• The domain of the function is the set of all real
numbers.
• Assign several values to the variable x and
compute the corresponding values for y

x y
3 10
2 5
1 2
0 1
1 2
2 5
3 10
64
Example Sketching a Graph
• Sketch the graph of the function defined by the
equation
• y x2 1
• Solution
• The domain of the function is the set of all real
numbers.
• Then plot these values in a graph

y
10 8 6 4 2
x y
3 10
2 5
1 2
0 1
1 2
2 5
3 10
x
3 2 1 1 2 3
65
Example Sketching a Graph
• Sketch the graph of the function defined by the
equation
• y x2 1
• Solution
• The domain of the function is the set of all real
numbers.
• And finally, connect the dots

y
10 8 6 4 2
x y
3 10
2 5
1 2
0 1
1 2
2 5
3 10
x
3 2 1 1 2 3
66
Example Sketching a Graph
• Sketch the graph of the function defined by the
equation
• Solution
• The function f is defined in a piecewise fashion
on the set of all real numbers.
• In the subdomain ( ?, 0), the rule for f is
given by
• In the subdomain 0, ?), the rule for f is given
by

67
Example Sketching a Graph
• Sketch the graph of the function defined by the
equation
• Solution
• Substituting negative values for x into
, while
• substituting zero and positive values into
we get

x y
3 3
2 2
1 1
0 0
1 1
2 1.41
3 1.73
68
Example Sketching a Graph
• Sketch the graph of the function defined by the
equation
• Solution
• Plotting these data and graphing we get

y
x y
3 3
2 2
1 1
0 0
1 1
2 1.41
3 1.73
3 2 1
x
3 2 1 1 2 3
69
The Vertical Line Test
• A curve in the xy-plane is the graph of a
function y f(x) if and only if each vertical
line intersects it in at most one point.

70
Examples
• Determine if the curve in the graph is a function
of x
• Solution
• The curve is indeed a function of x, because
there is one and only one value of y for any
given value of x.

y
x
71
Examples
• Determine if the curve in the graph is a function
of x
• Solution
• The curve is not a function of x, because there
is more than one value of y for some values of x.

y
x
72
Examples
• Determine if the curve in the graph is a function
of x
• Solution
• The curve is indeed a function of x, because
there is one and only one value of y for any
given value of x.

y
x
73
2.4
• The Algebra of Functions

74
The Sum, Difference, Product and Quotient
of Functions
• Consider the graph below
• R(t) denotes the federal government revenue at
any time t.
• S(t) denotes the federal government spending at
any time t.

y
2000 1800 1600 1400 1200 1000
y R(t)
y S(t)
S(t)
Billions of Dollars
R(t)
t
1990 1992 1994 1996 1998 2000
t
Year
75
The Sum, Difference, Product and Quotient
of Functions
• Consider the graph below
• The difference R(t) S(t) gives the budget
deficit (if negative) or surplus (if positive) in
billions of dollars at any time t.

y
2000 1800 1600 1400 1200 1000
y R(t)
y S(t)
S(t)
Billions of Dollars
D(t) R(t) S(t)
R(t)
t
1990 1992 1994 1996 1998 2000
t
Year
76
The Sum, Difference, Product and Quotient
of Functions
• The budget balance D(t) is shown below
• D(t) is also a function that denotes the federal
government deficit (surplus) at any time t.
• This function is the difference of the two
functions R and S.
• D(t) has the same domain as R(t) and S(t).

y
400 200 0 200 400
y D(t)
t
Billions of Dollars
t
1992 1994 1996 1998 2000
D(t)
Year
77
The Sum, Difference, Product and Quotient
of Functions
• Most functions are built up from other, generally
simpler functions.
• For example, we may view the function f(x) 2x
4 as the sum of the two functions g(x) 2x and
h(x) 4.

78
The Sum, Difference, Product and Quotient of
Functions
• Let f and g be functions with domains A and B,
respectively.
• The sum f g, the difference f g, and the
product fg of f and g are functions with domain A
n B and rule given by
• (f g)(x) f(x) g(x) Sum
• (f g)(x) f(x) g(x) Difference
• (fg)(x) f(x)g(x) Product
• The quotient f/g of f and g has domain A n B
excluding all numbers x such that g(x) 0 and
rule given by
• Quotient

79
Example
• Let and g(x) 2x 1.
• Find the sum s, the difference d, the product p,
and the quotient q of the functions f and g.
• Solution
• Since the domain of f is A 1,?) and the
domain of g is B ( ?, ?), we see
that the domain of s, d, and p is A
n B 1,?).
• The rules are as follows

80
Example
• Let and g(x) 2x 1.
• Find the sum s, the difference d, the product p,
and the quotient q of the functions f and g.
• Solution
• The domain of the quotient function is 1,?)
together with the restriction x ? ½ .
• Thus, the domain is 1, ½) U ( ½,?).
• The rule is as follows

81
Applied Example Cost Functions
• Suppose Puritron, a manufacturer of water
filters, has a monthly fixed cost of 10,000 and
a variable cost of
• 0.0001x2 10x (0 ? x ? 40,000)
• dollars, where x denotes the number of filters
manufactured per month.
• Find a function C that gives the total monthly
cost incurred by Puritron in the manufacture of x
filters.

82
Applied Example Cost Functions
• Solution
• Puritrons monthly fixed cost is always 10,000,
so it can be described by the constant function
• F(x) 10,000
• The variable cost can be described by the
function
• V(x) 0.0001x2 10x
• The total cost is the sum of the fixed cost F and
the variable cost V
• C(x) V(x) F(x)
• 0.0001x2 10x 10,000 (0 ? x ? 40,000)

83
Applied Example Cost Functions
• Lets now consider profits
• Suppose that the total revenue R realized by
Puritron from the sale of x water filters is
given by
• R(x) 0.0005x2 20x (0 x 40,000)
• Find
• The total profit function for Puritron.
• The total profit when Puritron produces 10,000
filters per month.

84
Applied Example Cost Functions
• Solution
• The total profit P realized by the firm is the
difference between the total revenue R and the
total cost C
• P(x) R(x) C(x)
• ( 0.0005x2 20x) ( 0.0001x2 10x
10,000)
• 0.0004x2 10x 10,000
• The total profit realized by Puritron when
producing 10,000 filters per month is
• P(x) 0.0004(10,000)2 10(10,000) 10,000
• 50,000
• or 50,000 per month.

85
The Composition of Two Functions
• Another way to build a function from other
functions is through a process known as the
composition of functions.
• Consider the functions f and g
• Evaluating the function g at the point f(x), we
find that
• This is an entirely new function, which we could
call h

86
The Composition of Two Functions
• Let f and g be functions.
• Then the composition of g and f is the function
ggf (read g circle f ) defined by
• (ggf )(x) g(f(x))
• The domain of ggf is the set of all x in the
domain of f such that f(x) lies in the domain of
g.

87
Example
• Let
• Find
• The rule for the composite function ggf.
• The rule for the composite function fgg.
• Solution
• To find ggf, evaluate the function g at f(x)
• To find fgg, evaluate the function f at g(x)

88
Applied Example Automobile Pollution
• An environmental impact study conducted for the
city of Oxnard indicates that, under existing
environmental protection laws, the level of
carbon monoxide (CO) present in the air due to
pollution from automobile exhaust will be
0.01x2/3 parts per million when the number of
motor vehicles is x thousand.
• A separate study conducted by a state government
agency estimates that t years from now the number
of motor vehicles in Oxnard will be 0.2t2 4t
64 thousand.
• Find
• An expression for the concentration of CO in the
air due to automobile exhaust t years from now.
• The level of concentration 5 years from now.

89
Applied Example Automobile Pollution
• Solution
• Part (a)
• The level of CO is described by the function
• g(x) 0.01x2/3
• where x is the number (in thousands) of motor
vehicles.
• In turn, the number (in thousands) of motor
vehicles is described by the function
• f(t) 0.2t2 4t 64
• where t is the number of years from now.
• Therefore, the concentration of CO due to
automobile exhaust t years from now is given by
• (ggf )(t) g(f(t)) 0.01(0.2t2 4t 64)2/3

90
Applied Example Automobile Pollution
• Solution
• Part (b)
• The level of CO five years from now is
• (ggf )(5) g(f(5)) 0.010.2(5)2 4(5)
642/3
• (0.01)892/3 0.20
• or approximately 0.20 parts per million.

91
2.5
• Linear Functions

92
Linear Function
• The function f defined by
• where m and b are constants, is called a linear
function.

93
Applied Example Linear Depreciation
• A Web server has an original value of 10,000 and
is to be depreciated linearly over 5 years with a
3000 scrap value.
• Find an expression giving the book value at the
end of year t.
• What will be the book value of the server at the
end of the second year?
• What is the rate of depreciation of the server?

94
Applied Example Linear Depreciation
• Solution
• Let V(t) denote the Web servers book value at
the end of the tth year. V is a linear function
of t.
• To find an equation of the straight line that
represents the depreciation, observe that V
10,000 when t 0 this tells us that the line
passes through the point (0, 10,000).
• Similarly, the condition that V 3000 when t 5
says that the line also passes through the point
(5, 3000).
• Thus, the slope of the line is given by

95
Applied Example Linear Depreciation
• Solution
• Using the point-slope form of the equation of a
line with point (0, 10,000) and slope m 1400,
we obtain the required expression
• The book value at the end of the second year is
given by
• or 7200.
• The rate of depreciation of the server is given
by the negative slope of the depreciation line m
1400, so the rate of depreciation is 1400 per
year.

96
Applied Example Linear Depreciation
• Solution
• The graph of V is

V
(0, 10,000)
10,000
(5, 3000)
3000
t
1 2 3 4 5 6
97
Cost, Revenue, and Profit Functions
• Let x denote the number of units of a product
manufactured or sold.
• Then, the total cost function is
• C(x) Total cost of manufacturing x units of
the product
• The revenue function is
• R(x) Total revenue realized from the sale of x
units of the product
• The profit function is
• P(x) Total profit realized from manufacturing
and selling x units of the product

98
Applied Example Profit Function
• Puritron, a manufacturer of water filters, has a
monthly fixed cost of 20,000, a production cost
of 20 per unit, and a selling price of 30 per
unit.
• Find the cost function, the revenue function, and
the profit function for Puritron.
• Solution
• Let x denote the number of units produced and
sold.
• Then,

99
Finding the Point of Intersection
• Suppose we are given two straight lines L1 and L2
with equations
• y m1x b1 and y m2x b2
• (where m1, b1, m2, and b2 are constants) that
intersect at the point P(x0, y0).
• The point P(x0, y0) lies on the line L1 and so
satisfies the equation y m1x b1.
• The point P(x0, y0) also lies on the line L2 and
so satisfies y m2x
b2 as well.
• Therefore, to find the point of intersection
P(x0, y0) of the lines L1 and L2, we solve for x
and y the system composed of the two equations
• y m1x b1 and y m2x b2

100
Example
• Find the point of intersection of the straight
lines that have equations
• y x 1 and y 2x 4
• Solution
• Substituting the value y as given in the first
equation into the second equation, we obtain
• Substituting this value of x into either one of
the given equations yields y 2.
• Therefore, the required point of intersection is
(1, 2).

101
Example
• Find the point of intersection of the straight
lines that have equations
• y x 1 and y 2x 4
• Solution
• The graph shows the point of intersection (1, 2)
of the two lines

y
5 4 3 2 1
L1
(1, 2)
x
1 1 2 3 4 5
L2
102
Applied Example Break-Even Level
• Prescott manufactures its products at a cost of
4 per unit and sells them for 10 per unit.
• If the firms fixed cost is 12,000 per month,
determine the firms break-even point.
• Solution
• The revenue function R and the cost function C
are given respectively by
• Setting R(x) C(x), we obtain

103
Applied Example Break-Even Level
• Prescott manufactures its products at a cost of
4 per unit and sells them for 10 per unit.
• If the firms fixed cost is 12,000 per month,
determine the firms break-even point.
• Solution
• Substituting x 2000 into R(x) 10x gives
• So, Prescotts break-even point is 2000 units of
the product, resulting in a break-even revenue of
20,000 per month.

104
2.6

105
• A quadratic function is one of the form
• where a, b, and c are constants and a ? 0.
• For example, the function
• is quadratic, with a 2, b 4, and c 3.

106
• Below is the graph of the quadratic function
• The graph of a quadratic function is a curve
called a parabola that opens upward or downward.

y
10 8 6 4 2
Parabola
x
2 1 1 2 3 4
107
• The parabola is symmetric with respect to a
vertical line called the axis of symmetry.
• The axis of symmetry also passes through the
lowest or highest point of the parabola, which is
called the vertex of the parabola.

Axis of symmetry
y
10 8 6 4 2
Parabola
Vertex (1, 1)
x
2 1 1 2 3 4
108
• We can use these properties to help us sketch the
• Suppose we want to sketch the graph of
• If we complete the square in x, we obtain
• Note that (x 1)2 is nonnegative it equals to
zero when x 1 and is greater than zero if x ?
1.
• Thus, we see that f(x) ? 2 for all values of
x.
• This tells us the vertex of the parabola is the
point (1, 2).

109
• We know the vertex of the parabola is the point
(1, 2) and that it is the minimum point of the
graph, since f(x) ? 2 for all values of x.
• Thus, the graph of f(x) 3x2 6x 1 looks as
follows

y
4 2 2
x
2 2 4
Vertex (1, 2)
110
• Given f(x) ax2 bx c (a ? 0)
• The domain of f is the set of all real numbers.
• If a gt 0, the parabola opens upward, and if a lt
0, it opens downward.
• The vertex of the parabola is
• The axis of symmetry of the parabola is
• The x-intercepts (if any) are found by solving
f(x) 0. The y-intercept is f(0)
c.

111
Example
• Given the quadratic function f(x) 2x2 5x
2
• Find the vertex of the parabola.
• Find the x-intercepts (if any) of the parabola.
• Sketch the parabola.
• Solution
• Here a 2, b 5, and c 2. therefore, the
x-coordinate of the vertex of the parabola is
• The y-coordinate of the vertex is therefore
given by
• Thus, the vertex of the parabola is the point

112
Example
• Given the quadratic function f(x) 2x2 5x
2
• Find the vertex of the parabola.
• Find the x-intercepts (if any) of the parabola.
• Sketch the parabola.
• Solution
• For the x-intercepts of the parabola, we solve
the equation
• using the quadratic formula with a 2, b 5,
and c 2.
• We find
• Thus, the x-intercepts of the parabola are 1/2
and 2.

113
Example
• Given the quadratic function f(x) 2x2 5x
2
• Find the vertex of the parabola.
• Find the x-intercepts (if any) of the parabola.
• Sketch the parabola.
• Solution
• The sketch

y
Vertex
1 1 2
x-intercepts
x
1 1 2
y-intercept
114
Some Economic Models
• Peoples decision on how much to demand or
purchase of a given product depends on the price
of the product
• The higher the price the less they want to buy of
it.
• A demand function p d(x) can be used to
describe this.

115
Some Economic Models
• Similarly, firms decision on how much to supply
or produce of a product depends on the price of
the product
• The higher the price, the more they want to
produce of it.
• A supply function p s(x) can be used to
describe this.

116
Some Economic Models
• The interaction between demand and supply will
ensure the market settles to a market
equilibrium
• This is the situation at which quantity demanded
equals quantity supplied.
• Graphically, this situation occurs when the
demand curve and the supply curve intersect
where d(x) s(x).

117
Applied Example Supply and Demand
• The demand function for a certain brand of
bluetooth wireless headset is given by
• The corresponding supply function is given by
• where p is the expressed in dollars and x is
measured in units of a thousand.
• Find the equilibrium quantity and price.

118
Applied Example Supply and Demand
• Solution
• We solve the following system of equations
• Substituting the second equation into the first
yields
• Thus, either x 400/9 (but this is not
possible), or x 20.
• So, the equilibrium quantity must be 20,000

119
Applied Example Supply and Demand
• Solution
• The equilibrium price is given by

120
2.7
Functions and Mathematical Models
121
Mathematical Models
• As we have seen, mathematics can be used to solve
real-world problems.
• We will now discuss a few more examples of
real-world phenomena, such as
• The solvency of the U.S. Social Security
trust fund
• Global warming

122
Mathematical Modeling
• Regardless of the field from which the real-world
problem is drawn, the problem is analyzed using a
process called mathematical modeling.
• The four steps in this process are

Real-world problem
Mathematical model
Formulate
Solve
Test
Solution of real-world problem
Solution of mathematical model
Interpret
123
Modeling With Polynomial Functions
• A polynomial function of degree n is a function
of the form
• where n is a nonnegative integer and the numbers
a0, a1, . an are
constants called the coefficients of the
polynomial function.
• Examples
• The function below is polynomial function of
degree 5

124
Modeling With Polynomial Functions
• A polynomial function of degree n is a function
of the form
• where n is a nonnegative integer and the numbers
a0, a1, . an are
constants called the coefficients of the
polynomial function.
• Examples
• The function below is polynomial function of
degree 3

125
Applied Example Global Warming
• The increase in carbon dioxide (CO2) in the
atmosphere is a major cause of global warming.
• Below is a table showing the average amount of
CO2, measured in parts per million volume (ppmv)
for various years from 1958 through 2007

Year 1958 1970 1974 1978 1985 1991 1998 2003 2007
Amount 315 325 330 335 345 355 365 375 380
126
Applied Example Global Warming
Year 1958 1970 1974 1978 1985 1991 1998 2003 2007
Amount 315 325 330 335 345 355 365 375 380
• Below is a scatter plot associated with these
data

y (ppmv)
380 360 340 320
t (years)
10 20 30 40 50
127
Applied Example Global Warming
Year 1958 1970 1974 1978 1985 1991 1998 2003 2007
Amount 315 325 330 335 345 355 365 375 380
• A mathematical model giving the approximate
amount of CO2 is given by

y (ppmv)
380 360 340 320
t (years)
10 20 30 40 50
128
Applied Example Global Warming
Year 1958 1970 1974 1978 1985 1991 1998 2003 2007
Amount 315 325 330 335 345 355 365 375 380
1. Use the model to estimate the average amount of
atmospheric CO2 in 1980 (t 23).
2. Assume that the trend continued and use the model
to predict the average amount of atmospheric CO2
in 2010.

129
Applied Example Global Warming
Year 1958 1970 1974 1978 1985 1991 1998 2003 2007
Amount 315 325 330 335 345 355 365 375 380
• Solution
• The average amount of atmospheric CO2 in 1980 is
given by
• or approximately 338 ppmv.
• Assuming that the trend will continue, the
average amount of atmospheric CO2 in 2010 will be

130
Applied Example Social Security Trust Fund
• The projected assets of the Social Security trust
fund (in trillions of dollars) from 2008 through
2040 are given by
• The scatter plot associated with these data is

Year 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2040
Assets 2.4 3.2 4.0 4.7 5.3 5.7 5.9 5.6 4.9 3.6 1.7 0
y (trillion)
6 4 2
t (years)
5 10 15 20 25 30
131
Applied Example Social Security Trust Fund
• The projected assets of the Social Security trust
fund (in trillions of dollars) from 2008 through
2040 are given by
• A mathematical model giving the approximate value
of assets in the trust fund (in trillions of
dollars) is

Year 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2040
Assets 2.4 3.2 4.0 4.7 5.3 5.7 5.9 5.6 4.9 3.6 1.7 0
y (trillion)
6 4 2
t (years)
5 10 15 20 25 30
132
Applied Example Social Security Trust Fund
Year 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2040
Assets 2.4 3.2 4.0 4.7 5.3 5.7 5.9 5.6 4.9 3.6 1.7 0
1. The first baby boomers will turn 65 in 2011. What
will be the assets of the Social Security trust
fund at that time?
2. The last of the baby boomers will turn 65 in
2029. What will the assets of the trust fund be
at the time?
3. Use the graph of function A(t) to estimate the
year in which the current Social Security system
will go broke.

133
Applied Example Social Security Trust Fund
Year 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2040
Assets 2.4 3.2 4.0 4.7 5.3 5.7 5.9 5.6 4.9 3.6 1.7 0
• Solution
• The assets of the Social Security fund in 2011 (t
3) will be
• or approximately 3.18 trillion.
• The assets of the Social Security fund in 2029
(t 21) will be
• or approximately 5.59 trillion.

134
Applied Example Social Security Trust Fund
Year 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2040
Assets 2.4 3.2 4.0 4.7 5.3 5.7 5.9 5.6 4.9 3.6 1.7 0
• Solution
• The graph shows that function A crosses the
t-axis at about t 32, suggesting the system
will go broke by 2040

y (trillion)
6 4 2
t (years)
5 10 15 20 25 30
135
Rational and Power Functions
• A rational function is simply the quotient of two
polynomials.
• In general, a rational function has the form
• where f(x) and g(x) are polynomial functions.
• Since the division by zero is not allowed, we
conclude that the domain of a rational function
is the set of all real numbers except the zeros
of g (the roots of the equation g(x) 0)

136
Rational and Power Functions
• Examples of rational functions

137
Rational and Power Functions
• Functions of the form
• where r is any real number, are called power
functions.
• We encountered examples of power functions
earlier in our work.
• Examples of power functions

138
Rational and Power Functions
• Many functions involve combinations of rational
and power functions.
• Examples

139
Applied Example Driving Costs
• A study of driving costs based on a 2007
medium-sized sedan found the following average
costs (car payments, gas, insurance, upkeep, and
depreciation), measured in cents per mile
• A mathematical model giving the average cost in
cents per mile is
• where x (in thousands) denotes the number of
miles the car is driven in 1 year.

Miles/year, x 5000 10,000 15,000 20,000
Cost/mile, y () 83.8 62.9 52.2 47.1
140
Applied Example Driving Costs
Miles/year, x 5000 10,000 15,000 20,000
Cost/mile, y () 83.8 62.9 52.2 47.1
• Below is the scatter plot associated with this
data

y ()
140 120 100 80 60 40 20
C(x)
x (?1000 miles/year)
5 10 15 20 25
141
Applied Example Driving Costs
Miles/year, x 5000 10,000 15,000 20,000
Cost/mile, y () 83.8 62.9 52.2 47.1
• Using this model, estimate the average cost of
driving a 2007 medium-sized sedan 8,000 miles per
year and 18,000 miles per year.
• Solution
• The average cost for driving a car 8,000 miles
per year is
• or approximately 68.8/mile.

142
Applied Example Driving Costs
Miles/year, x 5000 10,000 15,000 20,000
Cost/mile, y () 83.8 62.9 52.2 47.1
• Using this model, estimate the average cost of
driving a 2007 medium-sized sedan 8,000 miles per
year and 18,000 miles per year.
• Solution
• The average cost for driving a car 18,000 miles
per year is
• or approximately 48.95/mile.

143
Constructing Mathematical Models
• Some mathematical models can be constructed using
elementary geometric and algebraic arguments.
• Guidelines for constructing mathematical
models
• Assign a letter to each variable mentioned in the
problem. If appropriate, draw and label a figure.
• Find an expression for the quantity sought.
• Use the conditions given in the problem to write
the quantity sought as a function f of
one variable.
• Note any restrictions to be placed on the domain
of f by the nature of the
problem.

144
Applied Example Enclosing an Area
• The owner of the Rancho Los Feliz has 3000 yards
of fencing with which to enclose a rectangular
piece of grazing land along the straight portion
of a river.
• Fencing is not required along the river.
• Letting x denote the width of the rectangle, find
a function f in the variable x giving the area of
the grazing land if she uses all of the fencing.

145
Applied Example Enclosing an Area
• Solution
• This information was given
• The area of the rectangular grazing land is A
xy.
• The amount of fencing is 2x y which must equal
3000 (to use all the fencing), so
• 2x y 3000
• Solving for y we get
• y 3000 2x
• Substituting this value of y into the expression
for A gives
• A x(3000 2x) 3000x 2x2
• Finally, x and y represent distances, so they
must be nonnegative, so x ? 0 and y 3000 2x ?
0 (or x ? 1500).
• Thus, the required function is
• f(x) 3000x 2x2 (0 ? x ? 1500)

146
Applied Example Charter-Flight Revenue
flight, Leisure World Travel Agency charges 300
per person.
flight (assume this is the case), then each fare
is reduced by 1 for each additional person.
• Letting x denote the number of passengers above
200, find a function giving the revenue realized
by the company.

147
Applied Example Charter-Flight Revenue
• Solution
• This information was given.
• If there are x passengers above 200, then the
number of passengers signing up for the flight is
200 x.
• The fare will be (300 x) dollars per passenger.
• The revenue will be
• R (200 x)(300 x)
• x2 100x 60,000
• The quantities must be positive, so x ? 0 and 300
x ? 0 (or x ? 300).
• So the required function is
• f(x) x2 100x 60,000 (0 ? x ? 300)

148
End of Chapter