Problem. a Man's Boyhood Lasted of His Life, He Then Played Soccer for Of

CHAPTER 9

Rational Numbers, Real Numbers, and Algebra

1 Problem. A man’s boyhood lasted 6 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 a Q = a and b are integers, b = 0 . b| 6 n o 42 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Example. 3 7 3 6 0 0 , , , , , 4 2 5 4 6 4 Also, 1 13 3 45 3 = , 6 = 4 4 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 5 5 · = = 7 12 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# a a Consider = since ( a)( b) = ab. Also, b b a a a + a 0 + = = = 0, b b b b so a a = . b b a Theorem. Let 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 = 0 . b b b m 6 ⇣ a ⌘ (Additive Inverse) For every rational number , there exists a unique rational a b 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 47

3 3 ( 3)4 7(3) 24 ( 63) 12 21 12 + ( 21) 33 = = = = = 7 4 7(4) 108 28 28 28 12 21 33 33 = = = 28 28 28 28 33 inverse or opposite of |{z} 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 = 0 . b · b · b m 6 ⇣ 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 3 3 11 + = + = 0 = 0 21 · 7 21 7 21 7 7 21 · ⇣ ⌘ ⇣ ⌘ ⇣a ⌘⇣c ⌘ ⇣ ⌘h ⇣ ⌘i 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 49 Example. (1) 40 10 = 27 ÷ 9 40 ( 10) 4 ÷ = . 27 9 3 ÷ (2) 47 3 = 49 ÷ 49 47 47 = . 3 3 (3) 3 5 = 8 ÷ 6 3 3 ◆6◆ 9 = . ◆8◆ · 5 20 4 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 51 Second way: 5 10 11 5 = . Since 11 < 10 and 12 > 0, < . 6 12 12 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 ⇣ ⌘ 1 ⇣ 15⌘ 3 x > (Associative Property of Multiplication) 3 6 h⇣ 5⌘⇣ ⌘i 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. 7 2) 3x = 7 has no whole number solution, but is a fractional solution. 3 We extend to fractions. 5 3) 5x = 2 has no integer or fractional solution, but 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 Since x2 = 2, = 2, so = 2. Then a2 = 2b2, so a2 is even. b b2 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 whose square is 2. We huave shown this number to be irrational and represent it by p2. We can then use a compass to find its location on the real number line. Just as 4 has two square roots, 2 and 2, so does 2. We use p2 to indicate the positive or principal square root of 2 and p2 to indicate the other. Definition (Square Root). Let a be a nonnegative real number. Then the square root of a (i.e., the principal square root of a), written pa, is defined as pa = b where b2 = a and b 0. Since there are infinitely many primes p, and pp is always irrational (why?), there are infinitely many irrationals. 56 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Example. Without using a square root key, approximate p7 to the nearest thousandth. Since 22 = 4, (p7)2 = 7, and 32 = 9, so 2 < p7 < 3 . Since 2.52 = 6.25, 2.62 = 6.76, and 2.72 = 7.29, so 2.6 < p7 < 2.7 . Since 2.652 > 7, and 2.642 < 7, so 2.64 < p7 < 2.65 . Since 2.6452 < 7, and 2.6462 > 7, so 2.645 < p7 < 2.646 . So which do we take, 2.645 or 2.646? One more step should give us the answer. Since 2.64572 < 7, and 2.64582 > 7, so 2.6457 < p7 < 2.6468 . Since both of these round to 2.646, we say p7 2.646 to gthe nearest thousandth. ⇡

Note. For any nonnegative real number x, (px)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 Multiplication Closure Closure Commutative Commutative Associative Associative Identity (0) Identity (1) 1 Inverse ( a) Inverse (a for a = 0) Distributive of Multiplication over Addition 6 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. 1) If a 0, then pn a = b if and only if bn = a and b 0. 2) If a < 0 and n is odd, then pn a = b if and only if bn = a.

Note. 1) a is called the radicand and n the index. 2) pn a = b is read “the nth root of a,” and is call a radical. 3) For square roots, we ususally write pa instead of p2 a. n 4) pn a = a.

4 4 5) W e cannot take even roots of negative numbers, such as p 1, for if b = p 1 b4 = 1, which is impossible. The same is true for p 1. 58 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Example. (1) p4 256 = 4 since 44 = 256. (2) p4 256 = 4. (3) p3 27 = 3 since 33 = 27. (4) p6 64 = does not exist.

How to deﬁne rational exponents? Consider 51/2. We want 51/2 51/2 = 51 = 5. · But p5 p5 = 5. · So what about 51/2 = p5?

Definition (Unit fraction Exponent). Let a be any real number and n any positive integer. Then a1/n = pn a where 1) n is arbitrary when a 0, and 2) n must be odd when a < 0. 9.2. THE REAL NUMBERS 59 Example. 1) ( 64)1/3 = p3 64 = 4. 2) 321/5 = p5 32 = 2.

m Definition (Rational Exponents). Le a be a nonnegative number, and n a rational number in somplest form. Then m am/n = a1/n = (am)1/n. Example. 1) 4 ( 8)4/3 = ( 8)1/3 = ( 2)4 = 16, or ( 8)4/3 = ( 8)4 = 40961/3 = 16 ⇥ ⇤ Whixh way seems easier? ⇥ ⇤ 2) 42/3 = (42)1/3 = 161/3 = p3 16. 3) 4/3 1/3 4 4 1 1 64 = (64 ) = 4 = = . 44 256 Properties of Rational Exponents Let a and b represent positive real numbers, and m and n rational (not neces- sarily) positive exponents. Then aman = am+n ambm = (ab)m (am)n = amn m m n a m n a a = = a ÷ an 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, papb = a1/2 b1/2 = (ab)1/ = pab. · Example. Simplify p16 p48 so that the radicand is as small as possible. ⇥ p16 p48 = p16(p16 3) = (p16(p16p3) = (p16p16)p3 = 16p3. ⇥ · Example. Compute and simplify: p20 p5 + p45 = p4 5 p5 + p9 5 = · · p4p5 p5 + p9p5 = 2p5 p5 + 3p5 = (2 1 + 3) p5 = 4p5 | 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 61 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 7 2 x + = 4 2 5 3 2 2 7 x x = (transposing) 4 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 ⌘ 234 x = 5 62 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Solving Inequalities of the Form ax + b < cx + d. (< can be replaced by >, , , =)  6 234 Whereas the solution set of the previous problem is , solution sets for 5 inequalities are expressed ssimilar to x x < 5 . n o { | } Example.

4x + 5 7x 7 4x 7x + 5 7 3x + 5 7 3x 7 5 3x 12 1 1 ( 3x) ( 12) (note the direction change) 3  3 x 4  x x 4 { |  } Example. 3 5 1 x 2 < x + 2 6 3 3 5 1 x 2 x < 2 6 3 9 5 1 x x 2 < 6 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 7 x x < | 2 Example. Chad was the samen age as Shellyo , 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 Equa- tion). 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 1 4 (4x + 24) = 4 39 4 · h 4x + 24 =i 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) R for all a A, i.e., if every element is related to itself. 2 2 2) R is symmetric if whenever (a, b) R, then (b, a) R also, i.e., if a is related to b, then b is related to a. 2 2 3) R is transitive on A if whenever (a, b) R and (b, c) R, then (a, c) R, i.e., if a is related to b and b is related2to c, then a has2 related to c. 2 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 A, the function notation for the element in B assigned to a is f(!a), i.e., (a, f2(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 A is the range of f. The range is a subset of the codomain. { 2 } 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 cub! e. 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 + 40 = 45 · 20a = 5 1 |a = {z } 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 = 55F. 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 = 62.5F. 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 = 0. Graphs of quadratic functions are parabolas. Our function p is a quadratic6 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 = 0 and b > 0, but b = 1. 6 6 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%, com- pounded annually, and time t in years, the amount of principal after t years is given by the formula t P (t) = P0(1 + r) . 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 1 F (x) = x + 2. 2 h i 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 2 x < 6 , range = 1, 2, 3, 4 { |  } { } c) f(x) = 3 when 2 x < 4  f(x) = 1.5 when 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).