1) The document introduces the absolute value function and piecewise-defined functions. It contains examples investigating college tuition rates as a piecewise function and a stock trading rule involving absolute value. 2) Tables are used to represent piecewise functions and the absolute value of various inputs is calculated. Equations are written to represent piecewise functions and the meaning of absolute value is discussed. 3) The document emphasizes understanding rates of change and absolute value, especially in business applications. It stresses the importance of studying these concepts and notes that absolute value is always positive.

1. Mark Anthony G. Arrieta BSEd – Math – 4 Math 116A
Mr. Allen C. Barbaso Presentation 3
CHAPTER 4 Integers: Expanding a Mathematical System
4.3 The Absolute Value Function
Introduction:
 A student will be asked to lead a prayer.
 Recall the previous topic being discussed by asking a student.
 Introduce the purpose of studying the lesson.
 Ask the students about their idea on the new topic being presented.
Purpose:
1.) Introduce piecewise-defined functions.
2.) Introduce the notation for change in x, x.
3.) Investigate the absolute value function.
4.) Use the absolute value notation to state mathematical ideas, concepts and rules concisely.
Investigation:
College Tuition: The tuition policy at a large university allows students to take 12 or more
semester hours for a flat charge of $1425. The charge is $98 per semester hour if a student takes
less than 12 credit units hours and student fees and perks are not charged.
1.) Complete Table 1 for the given problem situation.
TABLE 1 Tuition Charges
Semester Hours Tuition ($)
6
9
11
12
15
18
2.) Table 1 displays a numeric representation of a function with input the number of semester
hours and output the tuition. State the domain of the problem situation.
3.) Sketch a graph of the function using your answers to Investigation 2 to generate pairs of
points on the graph. Clearly label the axes and the scale.
4.) Let h represent the number of semester hours and T represent the tuition charge.
a.) Write an algebraic representation (equation) for the problem situation if a student
takes fewer than 12 credit hours.
b.) Write an algebraic representation (equation) of the problem situation if a student takes
atleast 12 credit hours.

2. Discussion:
 The previous investigation revolved around a problem situation that required a split in the
domain. The function process depends on the input value. Such a function is called a
piecewise-defined function.
 The domain of the College Tuition problem must include all possible numbers for total
credit hours taken in a semester. This is the set of all whole numbers less than some arbitrary
upper limit. To write the function, we must split the domain. The computation of output
depends on whether the input is less than 12.
 If a student takes fewer than 12 credit hours, then the tuition charge is calculated by
multiplying $98 (the charge for one credit) by the number of credit hours taken. Using h as
the number of credit hours and T as the tuition charge, we can write
if h < 12 then T(h) = 98h.
 If students take 12 or more hours, they pay a flat rate of $1425. Using the same variables we
write
if h 12 then T(h) = 1425.
 This type of function requires a new type of function machine. The input must be evaluated
to determine which process to use. Upon entering the function machine, a decision about the
input is made. Based on the decision, we follow exactly one of the multiple paths.
Investigation:
Sarah’s Stock Portfolio. Sarah was a stockbroker who used a simple test to determine whether
to buy or sell a stock that was of interest. If the change in the stock from opening to closing
exceeded two points, she sold the stock if she owned it and bought the stock if she did not.
5.) Use the price of the stocks in Table 2 to decide which action (buy, sell, or wait) is determined
by Sarah’s rule. Sarah owns the stocks indicated by the asterisk (*).
TABLE 2 Sarah’s Stock Choices
Stock Open Close Change Action
*IBM 70 72
AMOCO 53 49
Apple 17 24
*Casio 20 18.5
Sharp 13 13
*TI 19 –3
HP 43 40.75
*SPC 11 7
DMW 62 3
Motorola 59 –2
Discussion:
 When computing an amount of change, the question of order of subtraction is an important
one. Since we are interested in change, we want to know the amount (magnitude) of change,
as well as the direction of the change. If the quantity gets larger, the change should be
positive, but if the quantity gets smaller, the change should be negative. This computation
can be accomplished by subtracting the initial value of the quantity from its final value.

3.  If p represents the price of a stock, let
p1 represent the initial value,
p2 represent the final value, and
p denote the change in p.
 The change in p is the final value minus the initial value, or p = p2 – p1.
Points to Ponder:
 The Greek letter delta, represented by the symbol , is used in mathematics for the phrase
change in. So the symbol p means the change between two values of p.
 Notice that we use small numbers below and to the right of the variable, such as 1 in p1.
These numbers, called subscripts, are used by mathematicians when they use the same
variable in several different settings.
 We are using the variable p to represent both the initial and final price. The subscripts 1 and
2 on p1 and p2 represent a symbolic way to distinguish between the two values. When you see
p1 and p2, read these as “p sub 1” and “p sub 2”. The word sub indicates subscript. In this
case, the subscript 1 is used to designate the initial value and the subscript 2 used to designate
the final value.
Investigation:
6.) a.) Complete Table 3 where the input is x and the output is the absolute value of x,
written as | |.
b.) For each answer in part (a), how does the output (answer) compare to the input (the
original value of x)?
TABLE 3 Absolute Value
x | |
5
–7
0
–3
8
2
–17
Discussion:
 As you noticed in the preceding investigation, unless you know where a quantity is positive
and where it is negative, you can’t determine the absolute value of that quantity. Absolute
value is a unary operation that has one input and one output. Before the output is determined,
we must know the sign of the input. If the input is positive or zero, then the output is the
same as the input. However, if the input is negative, then the output is the opposite of the
input.
Explorations:
1.) If x represents the money in my petty cash fund, what notation would be used to designate the
a.) amount of money at the beginning of the day (initial value)?
b.) amount of money at the end of the day (final value)?
c.) the change in cash during the day?

4. 2.) Complete Table 4 by finding x. Assume that x1 represents the initial value and x2 represents
the final value.
TABLE 4 Daily Change in Petty Cash Account
Initial Value x1 in $ Final Value x2 in $
Change in Petty
Cash x in $
5 3
3 5
3 3
0 7
7 0
0 –2
–2 0
–3 4
4 –3
1 –2
–3 3
3.) What are the solutions to the following equations?
a.) | | = 7
b.) | | = 0
c.) | | = –3
d.) | | = 11
Explorations: (Answer)
1.) If x represents the money in my petty cash fund, what notation would be used to designate the
a.) amount of money at the beginning of the day (initial value)?
Answer: x1
b.) amount of money at the end of the day (final value)?
Answer: x2
c.) the change in cash during the day?
Answer: x

5. 2.) Complete Table 4 by finding x. Assume that x1 represents the initial value and x2 represents
the final value.
TABLE 4 Daily Change in Petty Cash Account
Initial Value x1 in $ Final Value x2 in $
Change in Petty
Cash x in $
5 3 –2
3 5 2
3 3 0
0 7 7
7 0 –7
0 –2 –2
–2 0 2
–3 4 7
4 –3 –7
1 3 –2
–6 –3 3
3.) What are the solutions to the following equations?
a.) | | = 7 Solutions: –2 and 12
b.) | | = 0 Solutions: –5 only
c.) | | = –3 Solutions: No Solution
d.) | | = 11 Solutions: –8 and 14
Reflection:
Explaining the concept of absolute value is a great way to examine your skills and
knowledge on number system and integers. If a person will ask me what is an absolute value I
will just show an example using a number line for easy understanding. Also, I will emphasize
that absolute value is always positive and if that person tries to solve for the solution of an
absolute value equation and it happen that the right side equation is a negative number, therefore
that equation has no solution because as what I have said absolute value is always positive.
Understanding problems that involve the rate of change is necessary for most people especially
those works are in the field of business because a minimal amount of change has already a big
impact on their life. As a future math teacher, I will really point out the importance of studying
rate of change and absolute value including its absolute value notations.
Reference:
De Marois, Phil; McGowen, Mercedes and Whitkanack, Darlene (2001). “Mathematical
Investigations”. Liceo de Cayagan University, Main Library. Jason Jordan Publishing.