Name__________________ Period ____
Pre-Calculus
Unit One
Characteristics and
Applications of Functions
Enrichment Packet
Due: Thursday, September 8th (A-Day)
Friday, September 9th (B-Day)
1
Pre-Calculus with Mrs. Calhoun
Unit One: Characteristics and Applications of Functions
Date
Objective
Activities
EA = Enrichment Activity
DAY 1
A: 8/22
B: 8/23
DAY 2
A: 8/24
B: 8/25
DAY 3
A: 8/26
B: 8/29
DAY 4
Intro to Pre-Calculus and
Investigate Function
Characteristics
Explore Minimum and
Maximum Points, End
Behavior, and Even/Odd
Characteristics of Functions
Review Parent Functions and
Function Vocabulary
Using Functions to Solve
Real World Problems
A: 8/30
B: 8/31
DAY 5
A: 9/1
B: 9/2
Explore Piece-Wise
Functions
DAY 6
A: 9/6
B: 9/7
Examine Different Types of
Function Discontinuities
DAY 7
A: 9/8
B: 9/9
Review and Test Unit One
1. Classroom Expectations
2. Student Info
3. Card Game: Function
Descriptions
4. EA ”Parent Function
Checklist”
5. EA “Functions-Give One, Get
One”
1. Due: Parent Letter & Supply
2. EA “Function
Characteristics”
3. EA “Even & Odd”
4. EA “End Behavior”
5. Fly-Swatter game
1. EA “Function Vocabulary”
2. EA “Boost”
Homework
WS = Worksheet
Get parent letter signed &
graphing calculator /batteries –
due next class!
WS “Parent Function Checklist”,
“Function Practice” and “How
Many Hundreds? ” due …
DUE DATE:
A: Tuesday 8/30
B: Wednesday 8/31
Enroll in WebAssign and
Complete “Getting Started with
WebAssign - Mathematics”
DUE DATE:
A/B: Saturday 8/28
1. Due: “Parent Function
Checklist”, “Function Practice”
and “How Many Hundreds? ”
WS
2. Quiz Unit 1.1 “Function
Characteristics”
3. EA ”Text Message Mayhem”
1. EA “Piecewise - Defined
Functions”
3. Notes 2.1-2.2
Textbook or WebAssgin
1.Due: Page 155 & 167
2. EA ”Continuity”
3. Piece-Wise Project “Gone to
Pieces” Assigned
1. Unit One Review Sheet
2. Due: Enrichment Activity
Packet
3. Test: Unit One
Review Sheet Unit One
DUE DATE:
A: Tuesday 9/6
B: Wednesday 9/7
***All make-up homework and
quiz retakes must be completed
before Unit One test!!!***
Study for Unit One Test
“Gone to Pieces” Project due…
DUE DATE:
A: Monday 9/26
B: Tuesday 9/27
Tutoring Times:
Monday, Wednesday, Thursday & Friday 7:45 – 8:20 am
[email protected]
469-219-2180 ext 80437
2
3
Function Characteristics Review
4
How do I know if it is a function?
Types of functions studied in Algebra 2:
5
Function Characteristics
Domain:
Range:
Notations:
Algebraic: Use equality and inequality symbols
Examples: x > 6
Read as: _____________________________________________
y  2 Read as: _____________________________________________
Set: If the graph is discrete, list all of the values in squiggly brackets, separated by commas
Example: Domain = 1, 3, 5, 7, 9
Read as: _______________________________
If the graph is continuous, use squiggly brackets to define a variable and describe it algebraically.
Example: {𝑥|𝑥 ∈ 𝑅} Read as: ___________________________________________
 y  0
Read as: ___________________________________________
Interval: For each continuous portion of a graph, write the starting and ending point separated by a comma.
Example: 1, 5 Read as: __________________________________________
 0, 20
Read as: __________________________________________
 4,   Read as: __________________________________________
**** The ( symbol means the number is ________________, same as using a
The

symbol means the number is _______________, same as using a
on a number line
on a number line
Union: If two sets are being combined, use the  symbol which means “and” in mathematical sentences.
Example:  ,0    0,   Read as: __________________________________________
_________________________________________________________________________
6
Practice: Function Characteristics
A.) Tell the domain and range of the given function using both set notation and interval notation
B.) Find the coordinates of the maximum and minimum points. List the increasing and decreasing intervals.
Set Notation
1.)
Interval Notation
Domain
Range
Relative Maximum Point(s): _____________________________________________________________
Relative Minimum Point(s): ______________________________________________________________
Increasing Intervals: ____________________________________________________________________
Decreasing Intervals: ___________________________________________________________________
2.) Sketch the graph of f ( x) 
x2  1
x
Set Notation
Interval Notation
Domain
Range
Relative Maximum Point(s): _____________________________________________________________
Relative Minimum Point(s): ______________________________________________________________
Increasing Intervals: ____________________________________________________________________
Decreasing Intervals: ___________________________________________________________________
7
8
EVEN vs ODD Summary:
9
When looking at the extreme ends of the graph, you can describe the “end behavior” in one of three ways:
On the extreme LEFT, we say x   , and the end behavior is described as:
* f ( x)   , which means ______________________________________
* f ( x)  _____ , where a number is filled in for the _____, and means ________________
____________________________________________________________
OR
* f ( x)   , which means ______________________________________
On the extreme RIGHT, we say x   , and the end behavior is described as:
* f ( x)   , which means ______________________________________
* f ( x)  _____ , where a number is filled into the _____, and means ________________
____________________________________________________________
OR
* f ( x)   , which means ______________________________________
EXAMPLES:
Enter each function into a graphing calculator to determine its behavior on the extreme left (x   ) or right (x
  ) of the graph. Identify the end behavior (A, B, or C) exhibited by of each side of the graph of the given
function. If the end behavior approaches a numerical limit (option B), determine this numerical limit.
10
Practice: End Behavior
3.)
4.)
11
Function Applications
12
Tear this page OUT of you packet. Cut along the dotted lines, then glue each piece onto the Function Vocabulary page
where you think they should belong.
“goes down” from
left to right
relative “low point”
a function that has 180 0
rotational symmetry
with respect to the
origin (Or, it looks the
same right-side up as
it does upside-down.)
a function that has
reflectional symmetry
with respect to the
y-axis
whether the graph
“points up,” “points
down,” or “flattens out”
on the extreme left and
right of the graph
relative “high point”
“boundary line”
“goes up” from left
to right
13
MUST LEAVE BLANK
14
15
16
What is RANGE?
What is DOMAIN?
Introduction: Piece-Wise Defined Functions
17
18
Piecewise Functions
19
Piecewise Functions
20
Graphs and Attributes of Tricky Functions
I. Domain of functions with domain restrictions
When does a function have a domain restriction?
(A) When there is a variable in the denominator, how do I find the domain restriction?
(B) If the function is an EVEN numbered root, how do I find the domain restriction?
(a)
1
x 3
(b)
x2
x2 1
(c)
x 5
(d)
x
x  2x  8
2
II. Solve
A hotel chain charges $75 each night for the first two nights and $50 for each additional night’s
stay. The total cost T is a function of the number of nights, x, that a guest stays.
(a) Complete the expressions in the following piecewise defined function.

T ( x)  

, if 0  x  2
, if x > 2
(b) Find T(2), T(3), and T(5)
(c) What do your answers to part (b) represent?
21
Function Review
III. Sketch the graph of the piecewise defined functions:
3x  2, x  3
(a) f ( x)  
 x  4, x  3
1  x 2 , x  2
(b) f ( x)  
x2
 x,
IV. Determine whether the graph is a function of x:
V. Answer the following:
The graph of a function h is given.
(a) Find h(-2), h(0), and h(3).
(b) Find the domain and range of h.
22
Graph each function using a “decimal” window (Zoom #4) to observe the different ways in which
functions can lack continuity.
23
Continuity Examples
EXAMPLES:
1. Write the functions for each graph, determine where any discontinuities occur, classify each
discontinuity by type.
Functions:
Discontinuity
2. The scale in the bathroom only shows weights to the nearest pound. This means that if an object
weighs less than ½ of a pound, the scale will show 0, and if something weighs 4.48 pounds, the scale
will show the weight as 4. But if something weighs 3.51 pounds, the scale will show the weight as 4
pounds as well.
Complete the following chart and graph:
Actual Weight (x)
Weight shown on
Scale (y)
0 to 0.49
0.5 to ____
0
1
2
3
4
5
6
What type of discontinuity exists in this problem? _________________________
24
25