Download Chapter 9 - FacStaff Home Page for CBU

CHAPTER 9
Rational Numbers, Real Numbers, and Algebra
Problem. A man’s boyhood lasted 16 of his life, he then played soccer for
1
1
12 of his life, and he married after 8 more of his life. A daughter was born 9
years after his marriage, and her birth coincided with the halfway point of his
life. How old was the man when he died?
Strategy 14 – Solve an Equation.
This strategy may be appropriate when
• A variable has been introduced.
• The words is, is equal to, or equalsappear in a problem.
• The stated conditions can easily be represented with an equation.
Let m = the man’s age when he died.
1
1
1
1
m+ m+ m+9= m
6
12
8
2
Multiply each side by 24.
4m + 2m + 3m + 216 = 12m
216 = 3m
72 = m
The man was 72 when he died.
40
9.1. THE RATIONAL NUMBERS
41
9.1. The Rational Numbers
Where we are so far:
Each arrow represents “is a subset of.”
The rational numbers are an extension of both the Fractions and the Integers.
The need:
3
We can solve 5x = 3 using fractions to get x = , but what about 5x = 3?
5
3
We can solve x+7 = 0 using integers to get x = 7, but what about x+ = 0?
2
So we need to extend both the fractions and the integers to take care of these
problems.
Two approaches:
1) Focusing on the property that every number has an opposite, wew take the
fractions wioth their opposites as the rational numbers. Then we note the
12
7
integers are also included: 4 =
and 7 =
, for instance.
3
1
2) Focusing on the property that every nonzero number has a recipracal, we
form all possible “fractions” where the numerator is an integer and the
denominator is a nonzero integer.
We will follow the second approach:
Definition (Rational Numbers). The set of rational numbers is the set
na
o
Q=
|a and b are integers, b 6= 0 .
b
42
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Example.
3
,
4
7
,
2
Also,
3
3
,
5
1
13
=
,
4
4
6
,
4
6
0
,
6
0
4
3
45
=
7
7
Each fraction can be written in this form and each integer n can be written as
n
.
1
We now have:
Definition (Equality of Rational Numbers).
a
c
a c
Let and Then = if and only if ad = bc.
b
d
b d
Recall that “equal” refers to the abstract concept of quantity attached to a
number, while “equivalent” refers to the various numerals used for a number.
1 2 3
1
2
3
= = =
=
=
2 4 6
2
4
6
1
are all numerals representing the ration number .
2
2
2
2
6
12
2
=
=
=
=
=
3
3
3
9
18
3
2
are all numerals representing the rational number
.
3
9.1. THE RATIONAL NUMBERS
43
a
Theorem. Let be any rational number and n any nonzero integer.
b
Then
a an na
=
=
.
b
bn
nb
a
A rational number is in simplest form or lowest terms if a and b have no
b
common prime factors and b is positive.
Example.
3 5 0
,
,
are in lowest terms.
4
7 1
4
10
0
0
,
,
,
are not in lowest terms.
3
15
3 3
Example.
96
=
108
8 · 12
8
8
=
=
9 · 12
9
9
60
=
84
5 · 12
=
7 · 12
5 5
=
7 7
Definition (Addition of Rational Numbers).
a
c
Let and be any rational numbers. Then
b
d
a c ad + bc
+ =
.
b d
bd
Corollary.
a c a+c
+ =
.
b b
b
44
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Example.
3
5
( 3)12 + 8 · 5
36 + 40
4
1
+
=
=
=
=
8
12
8 · 12
96
96 24
"
#
9 10
1
=
+
=
24 24 24
Consider
so
a
a
=
since ( a)( b) = ab. Also,
b
b
a a
a+a 0
+ =
= = 0,
b
b
b
b
a
a
=
.
b
b
Theorem. Let
a
be any rational number. Then
b
a
a
a
=
=
.
b
b
b
Fractional Number Line:
9.1. THE RATIONAL NUMBERS
45
Properties of Rational Number Addition
a c
e
Let , , and be any rational numbers.
b d
f
a c
(Closure) + is a rational number.
b d
a c c a
(Commutative) + = + .
b d d b
⇣a c ⌘ e a ⇣ c e ⌘
(Associative)
+
+ = +
+
.
b d
f
b
d f
⇣
⌘
a
a
a
0
(Identity) + 0 = = 0 +
0 = , m 6= 0 .
b
b
b
m
a
(Additive Inverse) For every rational number , there exists a unique rational
b
a
number
such that
b
⇣ a⌘ a
a ⇣ a⌘
+
=0=
+ .
b
b
b
b
Example.
2 ⇣3 7⌘
+
+
=
(mentally?)
9
5 9
2 ⇣7 3⌘
+
+
=
9
9 5
⇣2 7⌘ 3
3
3 8
+
+ =1+ =1 =
9 9
5
5
5 5
3 ⇣ 18 17 ⌘
+
+
=
(mentally?)
11
66
23
3 ⇣ 18 17 ⌘
+
+
=
11
66
23
⇣3
3 ⌘ 17
17 17
+
+
=0+
=
11 11
23
23 23
46
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Theorem (Additive Cancellation).
a c
e
Let , , and be any rational numbers. Then
b d
f
a e
c e
a c
If + = + , then = .
b f
d f
b d
Theorem (Opposite of the Opposite).
a
Let be any rational number. Then
b
⇣ a⌘ a
= .
b
b
Definition (Subtraction of Rational Numbers).
a
c
Let and be any rational numbers. Then
b
d
a c a ⇣ c⌘
= +
.
b d b
d
Also,
a c a ⇣ c ⌘ a ⇣ c ⌘ a + ( c) a c
= +
= +
=
=
b b b
b
b
b
b
b
and
a c ad bc ad bc
=
=
.
b d bd bd
bd
Example.
2 ⇣ 7 ⌘ 2(12) 9( 7) 24 ( 63) 24 + 63
87
29
=
=
=
=
=
9
12
108
108
108
108 36
2
7
= +
9 12
8
21 29
=
+
=
36 36 36
9.1. THE RATIONAL NUMBERS
3
7
47
3 ( 3)4 7(3) 24 ( 63)
12 21
12 + ( 21)
33
=
=
=
=
=
4
7(4)
108
28
28
28
12 21
33
33
=
=
=
28
28
28
28
|{z}
33
inverse or opposite of
28
Definition (Multiplication of Rational Numbers).
a
c
Let and be any rational numbers. Then
b
d
a c ac
· = .
b d bd
Properties of Rational Number Multiplication
a c
e
Let , , and be any rational numbers.
b d
f
a c
(Closure) · is a rational number.
b d
a c c a
(Commutative) · = · .
b d d b
⇣a c ⌘ e a ⇣ c e ⌘
(Associative)
·
· = ·
·
.
b d f
b
d f
⇣
⌘
a
a
a
m
(Multiplicative Identity) · 1 = = 1 ·
1 = , m 6= 0 .
b
b
b
m
a
(Multiplicative Inverse) For every nonzero rational number , there exists a
b
b
unique rational number such that
a
a b
· = 1.
b a
Note. The multiplicative inverse of a rational number is also called the
reciprocal of the number.
48
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Distributive Property of Multiplication over Addition
a c
e
Let , , and be any rational numbers. Then
b d
f
a⇣ c e ⌘ a c a e
+
= · + · .
b d f
b d b f
Example.
⇣ 9 23 ⌘ ⇣ 7 ⌘
·
·
=
(mentally?)
7
27
9
⇣ 23
9⌘ ⇣ 7⌘
23 ⇣ 9
7⌘
23
23
23
·
·
=
·
·
=
·1=
=
27 7
9
27
7
9
27
27
27
3 ⇣ 11 ⌘ ⇣ 3 ⌘⇣ 11 ⌘
·
+
=
(mentally?)
7
21
7
21
⇣ 11 ⌘ ⇣ 3 ⌘ ⇣ 11 ⌘⇣ 3 ⌘ ⇣ 11 ⌘h 3 ⇣ 3 ⌘i
11
·
+
=
+
=
·0=0
21
7
21
7
21 7
7
21
a
c
c
Definition. Let and be any rational numbers where is nonzero.
b
d
d
Then
a c a d
÷ = ⇥ .
b d b c
Common Denominator Division:
a c a b a
÷ = ÷ = , so
b b b c c
a c a
÷ =
or a ÷ c.
b b c
⇣
a⌘
A third method since a ÷ b =
:
b
a c a d a d a d a c a÷c
÷ = · = · = · = ÷ =
,
b d b c c c c c
b d b÷d
so
a c a÷c
÷ =
.
b d b÷d
9.1. THE RATIONAL NUMBERS
Example.
(1)
(2)
40
10
÷
=
27
9
40 ÷ ( 10) 4
= .
27 ÷ 9
3
47
3
÷
=
49
49
47
47
=
.
3
3
(3)
3 5
÷ =
8
6
3
3 ◆6◆
9
· =
.
◆
8
5
20
◆
4
49
50
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Ordering Rational Numbers
Three equivalent ways (as we did with fractions):
1) Number-Line Approach:
a c ⇣
c a⌘
<
or >
if and only if
b d
d b
a
c
is to the left of on the rational number line.
b
d
2) Common-Positive-Denominator Approach:
a c
<
if and only if a < c and b > 0.
b c
3)Addition approach:
a c
p
< if and only if the is a positive rational number such that
b d
q
a p c
+ = , or equivalently,
b q d
a c
c a
< if and only if
is positive.
b d
d b
5
11
Example. Compare
and
.
6
12
First way:
11 ⇣ 5 ⌘
11 5
11 10
1
=
+ =
+
=
. Then
12
6
12 6
12 12
12
⇣ 5 ⌘ ⇣ 11 ⌘
1
11 ⇣ 5 ⌘
= , so
<
.
6
12
12
12
6
9.1. THE RATIONAL NUMBERS
Second way:
5
10
=
. Since
6
12
11 <
10 and 12 > 0,
11 ⇣
<
12
51
5⌘
.
6
Cross Multiplication Inequality
a
a
Let and be any rational numbers, where b > 0 and d > 0. Then
b
b
a c
< if and only if ad < bc.
b d
5
11
Example. Compare
and
.
6
12
Third way: Note that the denominators are positive. Since
11
5
( 5)12 = 60 and 6( 11) = 66, and
66 < 60,
<
.
12
6
Properties of Order for Rational Numbers
Transitive
Less than and addition
Less than and multiplication by a positive
Less than and multiplication by a negative
Density Property – between any two rational numbers there exists at least one
rational number
Similar Properties hold for >, , .
52
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
1
5
Example. Solve
x<
.
3
6
⇣ 1 ⌘
⇣ 5⌘
3
x > 3
(Multiplication by a negative)
3
6
h⇣
⌘⇣ 1 ⌘i
15
3
x>
(Associative Property of Multiplication)
3
6
5
1x >
(Inverse Property of Multiplication)
2
5
x>
(Identity Property of Multiplication)
2
9.2. The Real Numbers
We started with the whole numbers.
x
5 = 0 has 5 as a whole number solution.
But consider:
1) x + 5 = 0 has no whole number solution, but
5 is an integer solution.
We extend to the integers.
2) 3x = 7 has no whole number solution, but
7
is a fractional solution.
3
We extend to fractions.
3)
5x = 2 has no integer or fractional solution, but
5
is a rational solution.
2
We extend to rationals.
4) x2 = 2 has no rational solution, so we need to extend once again.
How do we know for sure that there is no rational solution to x2 = 2?
9.2. THE REAL NUMBERS
53
Theorem. There is no rational number whose square is 2.
Proof.
We use indirect reasoning.
Suppose x is a rational number whose square is 2.
a
Then x can be written in lowest terms as , where a is an integer and b is a
b
positive integer.
⇣ a ⌘2
a2
2
Since x = 2,
= 2, so 2 = 2. Then a2 = 2b2, so a2 is even.
b
b
But then a is even, so a = 2n for some integer n.
Then (2n)2 = 2b2, so 4n2 = 2b2.
Then 2n2 = b2, so b2 is even, and thus b is even.
Then a and b both have 2 as a common factor,
a
so cannot be in lowest terms, a contradiction.
b
Thus x cannot be rational.
⇤
We have learned that every fraction can be written as a repeating decimal, and
vice-versa. Then so can every rational number just by taking opposites.
Thus the irrational numbers, the numbers that are not rational, must have
infinite nonrepeating decimal representations.
Definition (The Real Numbers). The set of real numbers, R, is the set of
all numbers that have an infinite decimal representation.
54
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Example.
0.1010010001000010000010000001 . . . is irrational.
3.25736 =
3.25736736736 . . . is rational and nonterminating.
2.45 = 2.450 = 2.45000000 . . . is rational and terminating.
We now have:
The Real Number Line
The real numbers complete the number line, i.e., for each real number there is
a point on the line, and for each point on the line there is a unique real number.
The numbers can be ordered by using their infinite decimals.
9.2. THE REAL NUMBERS
55
Representing some real numbers geometrically.
Recall the Pythagorean Theorem. In a right triangle whose legs are lengths a
and b and whose hypotenuse (the long side) has length c,
a2 + b2 = c2.
12 + 12 = c2
1 + 1 = c2
2 = c2
Thus, the length of the hypotenuse c is the positive real number whosepsquare
is 2. We huave shown this number to be irrational and represent it by 2. We
can then use a compass to find its location on the real number line.
p
Just as 4 has two square roots, 2 and 2, sop
does 2. We use 2 to indicate the
positive or principal square root of 2 and
2 to indicate the other.
Definition (Square Root). Let a be a nonnegative real number.
Then the
p
square root of a (i.e., the principal square root of a), written a, is defined as
p
a = b where b2 = a and b 0.
Since there are infinitely many primes p, and
there are infinitely many irrationals.
p
p is always irrational (why?),
56
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
p
Example. Without using a square root key, approximate 7 to the nearest
thousandth.
p
p 2
2
2
Since 2 = 4, ( 7) = 7, and 3 = 9, so 2 < 7 < 3 .
p
2
2
2
Since 2.5 = 6.25, 2.6 = 6.76, and 2.7 = 7.29, so 2.6 < 7 < 2.7 .
p
Since 2.652 > 7, and 2.642 < 7, so 2.64 < 7 < 2.65 .
p
2
2
Since 2.645 < 7, and 2.646 > 7, so 2.645 < 7 < 2.646 .
So which do we take, 2.645 or 2.646?
One more step should give us the answer.
p
Since 2.6457 < 7, and 2.6458 > 7, so 2.6457 < 7 < 2.6468 .
2
2
Since both of these round to 2.646,
p
we say 7 ⇡ 2.646 to gthe nearest thousandth.
Note. For any nonnegative real number x,
p
( x)2 = x.
⇡ is also an irrational number. ⇡ is the ratio of the circumference to the diameter
22
of any circle. ⇡ ⇡
and ⇡ ⇡ 3.14159 (3.14 and 3.1416 are also often used).
7
Operation Properties of Real Numbers
Addition
Closure
Commutative
Associative
Identity (0)
Inverse ( a)
Distributive of Multiplication over Addition
Multiplication
Closure
Commutative
Associative
Identity (1)
Inverse ( a1 for a 6= 0)
9.2. THE REAL NUMBERS
57
Definition. For any two real numbers a and b, a < b if and only if there
exists a positive real number p such that a + p = b.
Ordering Properties of Real Numbers
(hold for <, >, )
Transitive
Less Than and Addition
Less Than and Multiplication by a Positive
Less Than and Multiplication by a Negative
Density
Rational Exponents
Definition (nth Root). Let a be a real number and n a positive integer.
p
1) If a 0, then n a = b if and only if bn = a and b 0.
p
2) If a < 0 and n is odd, then n a = b if and only if bn = a.
Note.
1) a is called the radicand and n the index.
p
2) n a = b is read “the nth root of a,” and is call a radical.
p
p
3) For square roots, we ususally write a instead of 2 a.
p n
4) n a = a.
p
p
5) We cannot take even roots of negative numbers, suchp
as 4 1, for if b = 4 1
b4 = 1, which is impossible. The same is true for
1.
58
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Example.
(1)
p
4
256 =
4 since 44 = 256.
(2)
p
4
256 =
4.
(3)
p
3
27 =
3 since
33 =
(4)
27.
p
6
64 =
does not exist.
How to define rational exponents?
Consider 51/2. We want 51/2 · 51/2 = 51 = 5.
p p
But 5 · 5 = 5.
p
So what about 51/2 = 5?
Definition (Unit fraction Exponent). Let a be any real number and n any
positive integer. Then
p
1/n
a = na
where
1) n is arbitrary when a
0, and
2) n must be odd when a < 0.
9.2. THE REAL NUMBERS
Example.
1/3
1) ( 64)
2) 321/5
p
3
=
64 =
p
= 5 32 = 2.
59
4.
m
Definition (Rational Exponents). Le a be a nonnegative number, and
n
a rational number in somplest form. Then
am/n = a1/n
m
= (am)1/n.
Example.
1)
⇥
⇤
1/3 4
( 8) = ( 8)
= ( 2)4 = 16, or
⇥
⇤
4/3
4
( 8) = ( 8) = 40961/3 = 16
4/3
Whixh way seems easier?
2)
42/3 = (42)1/3 = 161/3 =
3)
64
4/3
= (641/3)
4
=4
4
=
p
3
16.
1
1
=
.
44 256
Properties of Rational Exponents
Let a and b represent positive real numbers, and m and n rational (not necessarily) positive exponents. Then
aman = am+n
ambm = (ab)m
(am)n = amn
am
m
n
a ÷ a = n = am n
a
In advanced math one can also define real number exponents which follow the
same properties.
60
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Example. For a
0 and b 0,
p
p p
1/2
1/2
1/
a b = a · b = (ab) = ab.
p
p
Example. Simplify 16 ⇥ 48 so that the radicand is as small as possible.
p
p
p p
p p p
p p p
p
16 ⇥ 48 = 16( 16 · 3) = ( 16( 16 3) = ( 16 16) 3 = 16 3.
Example. Compute and simplify:
p
p
p
20
5 + 45 =
p
p
p
4·5
5+ 9·5=
p p
p
p p
4 5
5+ 9 5=
p
p
p
2 5
5+3 5=
p
p
(2 1 + 3)| 5 = 4 5
Algebra
Solving Equations of the Form ax + b = cx + d.
Method:
1) Add appropriate values to each side to obtain mx = n.
1
2) Multiply each side by
(or, equivalently, divide each side by m).
m
m
3) The solution is then x = .
n
9.2. THE REAL NUMBERS
Example.
4x + 12 =
3x + 8
4x + 12 + 3x = 3x + 8 +3x
7x + 12 = 8
In transposing a term from one side to the other, just change its sign.
7x = 8 12
7x = 4
4
4
x=
=
7
7
Example.
3
x
4
3
7
2
x+ =
4
2
5
2
2 7
x=
(transposing)
3
5 2
⇣3 2⌘
4
35
x=
+
4 3
10
10
⇣9
⌘
8
39
x=
12 12
10
1
39
x=
12
10
⇣
6
39 ⌘
⇢
x=⇢
12 ⇢
10
⇢
5
x=
234
5
61
62
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Solving Inequalities of the Form ax + b < cx + d.
(< can be replaced by >, , , 6=)
Whereas the solution set of the previous problem is
inequalities are expressed ssimilar to {x|x < 5}.
n
234 o
, solution sets for
5
Example.
4x + 5 7x 7
4x 7x + 5
7
3x + 5
7
3x
7 5
3x
12
1
( 3x) 
3
1
( 12)
3
(note the direction change)
x4
{x|x  4}
Example.
3
x
2
3
x
2
9
x
6
5
1
2< x+
6
3
5
1
2
x<
6
3
5
1
x 2<
6
3
4
1
x 2<
6
3
2
1
x< +2
3
3
2
7
x<
3
3
9.2. THE REAL NUMBERS
63
3⇣2 ⌘ 3⇣7⌘
x =
2 3
2 3
7
x<
2
n
7o
x|x <
2
Example. Chad was the same age as Shelly, and Holly was 4 years older
than both of them. Chad’s dad was 20 when Chad was born, and the average
age of the four of them is 39. What are their ages?
solution. Use strategy 2 (Use of Variable) and Strategy 14 (Solve an Equation).
Let x = Chad’s age = Shelly’s age
x + 4 = Holly’s age
x + 20 = Chad’s dad’s age
1
(sum of the ages) = 39
4
1
(x + x + x + 4 + x + 20) = 39
4
1
(4x + 24) = 39
4
h1
i
4 (4x + 24) = 4 · 39
4
4x + 24 = 156
4x = 156 24
4x = 132
1
1
(4x) = (132)
4
4
x = 33
Chad and Shelly are 33, Holly is 37, and Chad’s dad is 53.
⇤
64
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
9.3. Relations and Functions
Relations are the description of relationships between 2 sets.
Definition. A relation from a set A to a set B is a subset of A ⇥ B. If
A = B, we say R is a relation on A.
Example. Let A = {1, 2, 3, 4, 5, 6} and the relation be “has the same number of factors as.”
or
R = {(1, 1), (2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (4, 4), (5, 2), (5, 3), (5, 5), (6, 6)}
For a relation R on a set A:
1) R is reflexive if (a, a) 2 R for all a 2 A, i.e., if every element is related to
itself.
2) R is symmetric if whenever (a, b) 2 R, then (b, a) 2 R also, i.e., if a is
related to b, then b is related to a.
3) R is transitive on A if whenever (a, b) 2 R and (b, c) 2 R, then (a, c) 2 R,
i.e., if a is related to b and b is related to c, then a has related to c.
R is an equivalence relation on a set A if it is reflexive, symmetric, and transitive.
Example. The relation R is the previous example is an equivalence relation.
9.3. RELATIONS AND FUNCTIONS
65
An equivalence relation creates a partition of the set A as a collection of
nonempty, pairwise disjoint sets whose union is A.
Such a partition can also be used to define an equivalence relation by using a
is related to b if they are in the same subset.
Example. The set A = {1, 2, 3, 4, 5, 6, 7, 8} with the relation “has more
factors than” is transitive, but not reflexive or symmetric. The relation “is not
equal to” on the same set is symmetic, but not reflexive or transitive.
Definition. A function is a relation that matches each element of a first
set to an element of a second set in such a way that no element in the first set
is assigned to two di↵erent elements in the second set, i.e., is a relation where
no two ordered pairs have the same first element.
A function f that assigns an element of a set A to an element of set B is written
f : A ! B. If a 2 A, the function notation for the element in B assigned to
a is f (a), i.e., (a, f (a)) is an ordered pair of the function (also relation) f . A
is the domain of f and B the codomain of f . The set {f (a) : a 2 A} is the
range of f . The range is a subset of the codomain.
Example.
1) None of our previous examples of relations were functions.
2) A sequence, a list of numbers arranged in order, called terms, is a function
whose domain is the set of whole numbers. “1” is matched with the first or
initial term, “2” with the second term, etc.
66
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
3, 5, 7, 9, 11, . . .
is an arithmetic sequence with initial term 3 and common di↵erence 2, i.e.,
successive terms di↵er by 2. The form is
a, a + d, a + 2d, a + 3d, . . . .
2, 6, 18, 54, 162, . . .
is a geometric sequence with initial term 2 and common ratio 3, the number
each successive term is multiplied by. The form is
a, ar, ar2, ar3, . . . .
3) f : W ! W defined by f (n) = n3 assigns each whole number (the domain)
to its cube.
Problem (Page 408 # 8).
a) is a function for a given election, but may not be in general.
b) not a function since many large cities, such as Memphis, have seral zip codes.
c) is a function (if I understand my biology).
d) is not a function since a person may have more than one pet.
9.4. FUNCTIONS AND THEIR GRAPHS
67
9.4. Functions and Their Graphs
Review – the Cartesian Coordinate System.
Note.
1) l and m are usually x and y for the x- and y-axes. Any letters or names
may be used.
2) The Roman numerals indicate quadrants. The quadrants do not include the
axes.
3) Each point in the plane is represented by an ordered pair of numbers (x, y),
the coordinates of the point.
The x-coordinate of a point is the perpendicular distance of the point from
the y-axis.
The y-coordinate of a point is the perpendicular distance of the point from
the x-axis.
4) The origin is the point (0, 0).
5) We say we have a coordinate system.
68
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Graphs of Linear Functions
Linear Functions are functions whose graphs are lines.
Recall: A function is a relation each element of a first set, called the domain,
to an element of a second set, called the range,in such a way that no element
of the first set is assigned to more than one element in the second set.
Example. The following table displays the number of cricket chirps per
minute at various temperatures:
cricket chirps per minute, n 20 40 60 80 100
temperature, T ( F)
45 50 55 60 65
We graph the points (n, T ):
We note the points (n, T ) fall on a line. Thus we have the graph of a linear
function. A linear function has the form
f (x) = ax + b,
9.4. FUNCTIONS AND THEIR GRAPHS
69
where a and b are constants. For our example, the formula would be
T (n) = an + b.
Can you find a and b? (Hint: Find b first!)
Look at the leftward extension of the graph to n = 0.
It looks as though T = 40.
If T (0) = a · 0 + b = b, then b = 40.
So we have T (n) = an + 40.
To find a, we try any other point on the graph, such as (20, 45) or T (20) = 45.
Then
T (20) = a
| · 20 +{z40 = 45}
20a = 5
1
a=
4
1
Our formula is thus T (n) = n + 40.
4
if you check, this formula works for all 5 of our points. We’ll just look at (60, 55).
1
T (60) = · 60 + 40 = 15 + 40 = 55 F.
4
We can also use the formula to predict other values, such as the temperature
when a cricket chirps 90 times per minute.
1
T (90) = · 90 + 40 = 22.5 + 40 = 62.5 F.
4
What is the domain here, the possible values of n?
n must be:
1) nonnegative.
2) a whole number.
3) not too large (for biologists to determine).
70
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Graphs of Quadratic Functions
Example. A ball is tossed up vertically at a velocity of 72 feet per second
from a point 10 feet above the ground. It is known from physics that the height
of the ball above the ground, in feet, is given by the position function
p(t) =
16t2 + 72t + 10,
where t is the time in seconds. At what time t is the ball at its highest point?
solution.
A quadratic function is a function of the form f (x) = ax2 + bx + c where a, b,
and c are constants and a 6= 0. Graphs of quadratic functions are parabolas.
Our function p is a quadratic function. We form a table of values and its graph.
t (sec) 0 1 2 3 4
p(t) (ft) 10 66 90 82 42
We plot our points and draw a smooth curve through them. The ball is at its
highest point between 2 and 3 seconds.
9.4. FUNCTIONS AND THEIR GRAPHS
71
We can use a graphing calculator to find the highest point. We can also find
that f (4.5) = 10. Since we have two t-coordinates with the same height, 10, the
time of the highest point is halfway between 0 and 4.5, namely t = 2.25.
⇤
Graphs of Exponential Functions
An exponential function is of the form f (x) = abx where a 6= 0 and b > 0, but
b 6= 1.
Example. How long does it take to double your money when the interest
rate is 2% compunded annually? Assume $100.00 is deposited.
solution.
It is known that for an initial principal P0 and an interest rate of 100r%, compounded annually, and time t in years, the amount of principal after t years is
given by the formula
P (t) = P0(1 + r)t.
For our example, the formula is
72
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
P (t) = 100(1 + .02)t = 100(1.02)t.
The graph is below in red.
To answer the question asked, we draw a horizontal line (blue) at P (t)=200.
Where this line meets the graph, we draw a vertical line to the t-axis. It looks as
though this line meets the t-axis just to the right of 35. We find P (35) = 199.99
and P (36) = 203.99.
Keeping in mind that interest is only paid at the end of a time period, it will
take 36 years to (at least) double our money.
⇤
9.4. FUNCTIONS AND THEIR GRAPHS
73
Step Functions
The graph pictured below is a graph of a step function, since its values are
pictured in a series of line segments or steps. A postage function is another
example of a step function.
If the greatest integer function
f (x) = [x]
is defined to be the greatest integer less than or equal to x, the formula for the
above graph is
h1 i
F (x) = x + 2.
2
a) f ( 1) =
1
f (2) =
3
f (3.75) =
3
74
9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
b) What are the domain and range?
domain = {x|
c) f (x) = 3 when
f (x) = 1.5 when
2  x < 6},
range = {1, 2, 3, 4}
2x<4
never occurs
Example. Water is poured at a constant rate into the three containers
shown below. Which graph corresponds to which container?
The height would rise at a constant rate in (b), which means a graph with a
constant slope. This is (i).
In (a), the water rises quickly at first due to the narrow bottom, but then slows
down. This would be like graph (ii).
Since (c) goes from wider to narrower, the water rises slowly at first, but then
faster as time goes on. This is graph (iii).