IG Unit 5 Parking Deck Pandemonium Learning Task 3
Name ___________________
Date____________
In this task, you will explore a particular type of piecewise function called a step
function. Although there are many different kinds of step functions, two common ones
are the least integer function, or the “ceiling function,” and the greatest integer
function, sometimes called the “floor function.”
The fee schedule at parking decks is often modeling using a step function. Let’s look at a
few different parking deck rates to see the step functions in action. (Most parking decks
have a maximum daily fee. However, for our exploration, we will assume that this
maximum does not exist.)
1. As you drive through town, Pete’s Parking Deck advertises free parking up to the first
hour. Then, the cost is $1 for each additional hour or part of an hour.
Thus, if you park at Pete’s Parking Deck for 59 minutes and 59 seconds, parking is
free; however, if the time shows at exactly 60 minutes, you pay $1. Similarly, if you
park for any time from 1 hour up to 2 hours, then you owe $1; but parking for exactly
2 hours costs $2.
a. Make a table listing some fees for parking at Pete’s for positive times that are 5
hours or less. Be sure to include some non-integer values; write these in decimal
form. Then draw (by hand) the graph that illustrates the fee schedule at Pete’s
for x hours, where 0  x  5 .
b. Use your graph to determine the fee if you park for 3 ½ hours. What about 3
hours, 55 minutes? 4 hours, 5 minutes?
c. What are the x- and y-intercepts of this graph? What is the interpretation in the
context of Pete’s Parking Deck?
d. What do you notice about the time, written in decimal form, and the
corresponding fee? Make a conjecture about the fee if you were to park at Pete’s
Parking Deck for 10.5 hours (assuming no maximum fee).
e. Write a piecewise function P to model the fee schedule at Pete’s Parking Deck.
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IG Unit 5 Parking Deck Pandemonium Learning Task 3
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2. If Pete’s Parking Deck allows fees to accumulate for multiple days for a car that is
just left in the lot, then, theoretically, there is no maximum fee. Thus, to write the
rule for your piecewise function model in Item 1, part d, immediately above, the
statement of the rule for P needed to show a pattern that continues forever.
There is a useful standard function that gives the same values as the function for the
parking fees at Pete’s Parking Deck but is defined for negative real numbers as well
nonnegative ones. This function is called the greatest integer function. The greatest
integer function is determined by locating the greatest integer that is less than or
equal to the x-value in question. For any real number x,  x  , is used to denote the
greatest integer function applied to x.
a. Evaluate each of the following by determining the greatest integer less than or
equal to the x-value, that is, let f x   x , where x is any real number.
i) f  3.6
ii) f  0.4 
iii) f  0.4 
iv) f  1
v) f  2.2 
b. Draw the graph of the greatest integer function, f x   x , for the viewing
window  10  x  10 .
c. What is the domain of the greatest integer function, f x   x ?
d. What is the range of the greatest integer function, f x   x ?
e. What is the shape of the graph beyond the given viewing window? Can you
indicate this on your hand-drawn graph?
Several different notations are used for the greatest integer function. The two most
common are f x   x , which we have used so far, and f x    x . However,
computer scientists use another name for the greatest integer function; they call it the
floor function, and use the notation f x   x  . To help remember this notation, note
that the bars on the brackets occur only at the bottom (or floor) of the straight line
segments.
The name “floor function” may be more helpful in remembering how the formula for the
function works. This function, by whatever name it is called, gives an the integer value
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IG Unit 5 Parking Deck Pandemonium Learning Task 3
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Date____________
output that is less than or equal to the value of the input number. Of course, there are
many integers less than or equal to any given number, so to make this a function, we
choose the largest integer that meets this condition. Choosing the largest integer that is
less than or equal to the input number gives us the name “greatest integer function” but
that name can be misleading because the output is always less than or equal to the input.
The name “floor function” should help you remember that the output is less than or equal
to the input number just as the height of the floor of a room is less than or equal to the
height of any object in the room.
3. Practice working with the various notations for the greatest integer function.
a. For each expression below, rewrite the expression using each of the other
notations.
i) 5.3
ii) 4.317
iii) 10.1  3.4
iv) 2.3  5.7
v)
(1.34)(6.8) 
b. Evaluate each expression in part a.
4. Paula’s Parking Deck is down the street from Pete’s. Paula recently renovated her
deck to make the parking spaces larger, so she charges more per hour than Pete.
Paula’s Parking Deck offers free parking up to the first hour (i.e., the first 59
minutes). Then, the cost is $2 for each additional hour or part of an hour. (If you park
for 1 ½ hours, you owe $2.)
a. Draw the graph that illustrates the fee schedule at Paula’s Parking Deck for x
hours, where 0  x  5 .
b. How does the graph for Paula’s Parking Deck compare with the graph of Pete’s
Parking Deck (from Item 1, part a)? To what graphical transformation does this
change correspond?
c. If you were to form a line by connecting the left endpoints of the steps in the
graph for Pete’s Parking Deck, found in answering Item 1, part a, what would be
the equation of the resulting linear function?
d. If you were to form a line by connecting the left endpoints of the steps in the
graph for Paula’s Parking Deck, found in answering part a for this item (Item 4),
what would be the equation of the resulting linear function?
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IG Unit 5 Parking Deck Pandemonium Learning Task 3
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Date____________
e. How do your answers for parts c and d relate to your answer to part b in this item
(Item 4)?
f. Write the function, g, in terms of the greatest integer function, that gives the same
values as the function for the parking fees at Paula’s Parking Deck but extends the
domain to include all real numbers.
g. Draw the graph of y  g x  over the domain  10  x  10 .
h. What are the domain and range for the function g?
5. Pablo’s Parking Deck is across the street from Paula’s deck. Pablo decided not to
provide any free parking. Pablo charges $1 for less than an hour, $2 for an hour or
more but less than two hours, and so forth, adding $1 whenever the time goes over the
next hour mark. (If you park for 59 minutes and 59 seconds, you owe $1; if you park
for 1 hour, you owe $2; etc.)
a. Draw the graph that illustrates the fee schedule at Pablo’s Parking Deck for x
hours, where 0  x  5 .
b. How does the graph for Pablo’s Parking Deck compare with the graph for Pete’s
Parking Deck? To what graphical transformation does this change correspond?
c. Write the function, h, in terms of the greatest integer function, that gives the same
values as the function for the parking fees at Pablo’s Parking Deck but extends the
domain to include all real numbers. (What are the two different forms that this
function could take?)
d. Draw the graph of y  hx over the domain  10  x  10 .
e. What are the domain and range for the function h?
6. Padma’s Parking Deck is the last deck on the street. To be a bit more competitive,
Padma decided to offer parking for each full hour at $1/hour. (If you park for 59
minutes or exactly 1 hour, you owe $1; if you park for up to and including 2 hours,
you owe $2.)
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IG Unit 5 Parking Deck Pandemonium Learning Task 3
Name ___________________
Date____________
a. Draw the graph that illustrates the fee schedule at Padma’s Parking Deck for x
hours, where 0  x  5 .
b. To which of the graphs of the other parking deck rates is the graph for Padma’s
Parking Deck most similar? How are the graphs similar? How are they different?
The fee schedule at Padma’s Parking Deck is modeled by the least integer function, or
ceiling function. The least integer function is determined by locating the least integer
that is greater than or equal to the x-value in question. The least integer functions is
also called the ceiling function and written with the following notation (analogous to the
floor function notation): c  x    x  . To help remember this notation, note that the bars
on the brackets occur only at the top (or ceiling) of the straight line segments.
7. Let c  x    x  .
a. Evaluate each of the following by determining the least integer greater than or
equal to the x-value.
i) c  3.5
ii) c  4 
iii) c  2.1 iv) c  .1
v) c 1.6 
b. Draw the graph of y  c  x  over the domain  10  x  10 .
c. What are the domain and range for the function c?
d. Suppose Padma chose to offer the first full hour free. After that, patrons would be
charged $1 for up through each full hour. What transformation of the least integer
function would model this parking fee `tructure?
8. As additional practice with step functions, graph each of the following.
For each function, state the parent function (either f x   x or g x   x ) and
explain what transformations have been applied to the parent function; state domain,
range, and y-intercept.
a. h  x   2  x  1
b. j x   x  2
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c. k x   x  2