Unit 7, Ongoing Activity, Little Black Book of Algebra II Properties
Little Black Book of Algebra II Properties
Unit 7 – Advanced Functions
7.1
7.2
7.3
1
x
, x , log x, 2 .
x
Continuity – provide an informal definition and give examples of continuous and discontinuous
functions.
Increasing, Decreasing, and Constant Functions – write definitions and draw example graphs such
Basic Graphs  graph and locate f(1): y = x, x2, x3,
x,
3
x, x ,
as y  9  x ; state the intervals on which the graphs are increasing and decreasing.
Even and Odd Functions – write definitions and give examples, illustrate properties of symmetry,
and explain how to prove that a function is even or odd (e.g., prove that y = x4 + x2 + 2 is even and
y = x3 + x is odd).
General Piecewise Function – write the definition and then graph, find the domain and range, and
2
7.4
7.5
solve the following example: f ( x ) 
2x  1
R
S
T x
2
if x  5
if x  5
for f (4) and f (1).
For properties 7.6  7.9 below, do the following:
 Explain in words the effect on the graph.
 Give an example of the graph of a given abstract function, and then the function transformed (do
not use y = x as your example).
 Explain in words the effect on the domain and range of a given function. Use the domain [–2, 6]
and the range [–8, 4] to find the new domain and range of the transformed function.
7.6 Translations (x + k) and (x  k), (x) + k and (x)  k
7.7 Rotations (–x) and –(x)
7.8 Dilations (kx), (|k|<1 and |k|>1), k(x) (|k|<1 and |k|>1)
7.9 Reflections and Rotations (|x|) and |(x)|
Blackline Masters, Algebra II
Page 7-1
Unit 7, Activity 1, Math Log Bellringer
Algebra II  Date
Graph the following by hand
and locate the zeroes and f(1).
(1) f(x) = x
(6) f(x) = 2x
1
f(x) =
2
(2) f(x) = x
(7)
x
3
f
x
=
x
f
(
x
)

x


(3)
(8)
(4) f(x) = x3
(5) f(x) = |x|
Blackline Masters, Algebra II
(9) f(x) = log x
(10)f  x  = x
Page 7-2
Unit 7, Activity 1, Vocabulary Card Template
Back of Vocabulary Card:
Blackline Masters, Algebra II
Page 7-3
Unit 7, Activity 2, Translations
Name
Date
Abstract Shifts:
Use the abstract graphs of g(x) below to answer questions #1  5.
(1) What is the domain of g(x)?
range?
Draw the graph of the following over the graph of g(x), label the new points, and find the new
domain and range:
(4, 8)
(4, 8)
g(x)
4
g(x)
4
(1, 2)
(1, 2)
(–5, –3)
(–5, –3)
(2) g(x) + 3 D:
(3) g(x) – 3 D:
R:
R:
(4, 8)
(4, 8)
g(x)
4
(1, 2)
(–5, –3)
(4) g(x + 3)
D:
g(x)
4
(1, 2)
(–5, –3)
R:
(5) g(x – 3) D:
R:
Practice without a graph: If the domain of f(x) is [4, 10] and the range is [6, 5] find the
domains and ranges of the following. If they do not change, write “same.”
(6) f(x  8)
D:
R:
Blackline Masters, Algebra II
(7) f(x)  8 D:
R:
Page 7-4
Unit 7, Activity 2, Translations
Parent Function Shifts:
State the parent function f(x) and the domain and range of the parent function. Graph the parent
function and the shifted function by hand and state the new domain and range.
(8) j  x   x  2  3 parent f(x) =
f(x)
j(x)
D:
D:
(10) h( x)  x  5  7
f(x) D:
h(x) D:
R:
R:
parent f(x) =
R:
R:
Blackline Masters, Algebra II
1
 2 parent f(x) =
x 3
D:
R:
D:
R:
(9) k  x  
f(x)
k(x)
(11) t(x) = log(x + 4) +3 parent f(x) =
f(x) D:
R:
t(x) D:
R:
Page 7-5
Unit 7, Activity 2, Translations with Answers
Key
Name
Date
Abstract Shifts:
Use the abstract graphs of g(x) below to answer questions #1  5.
(1) What is the domain of g(x)?
[5, 4]
range?
[3, 8]
Draw the graph of the following over the graph of g(x), label the new points, and find the new
domain and range:
(4, 11)
7
(4, 8)
(4, 8)
(1, 5)
4
g(x)
g(x)
(4, 5)
4
(1, 2)
(1, 2)
1
(–5, 0)
(–5, –3)
(1, 1)
(–5, –3)
(5, 6)
(2) g(x) + 3 D:
[5, 4] R:
[0, 11]
(3) g(x) – 3 D:
[5, 4]
(1, 8) (4, 8)
(2, 2)
(8, 3)
(7, 8)
g(x)
4
(1, 2)
(1, 2)
(–5, –3)
(4) g(x + 3)
(4, 8)
g(x)
4
[6, 5]
R:
(4, 2)
(–5, –3)
(-2, 3)
D:
[8, 1] R:
[3, 8]
(5) g(x – 3) D:
[-2, 7]
R:
[3, 8]
Practice without a graph: If the domain of f(x) is [4, 10] and the range is [6, 5], find the
domains and ranges of the following. If they do not change, write “same”:
(6) f(x  8)
D:
[4, 18] R:
Blackline Masters, Algebra II
same
(7) f(x)  8 D: same
R:
[14, 3]
Page 7-6
Unit 7, Activity 2, Translations with Answers
Parent Function Shifts:
State the parent function f(x) and the domain and range of the parent function. Graph the parent
function and the shifted function by hand and state the new domain and range.
(8)
j  x   x  2  3 parent f(x) =
f(x)
j(x)
D:
D:
[0, ) R: [0. )
[2, ) R: [3, )
(10) h( x)  x  5  7
f(x)
h(x)
x
1
1
 2 parent f(x) =
x 3
x
D: x ≠ 0
R:
y≠0
D: x ≠ 3
R:
y≠2
(9) k  x  
f(x)
k(x)
parent f(x) =
D: all reals R: [0, ∞)
D: all reals R: [7, ∞)
Blackline Masters, Algebra II
(11) t(x) = log(x + 4) +3 parent f(x) =
f(x)
t(x)
D: (0, ∞)
D: (4, ∞)
R: all reals
R: all reals
Page 7-7
Unit 7, Activity 3, Rotations Discovery Worksheet
Name
Date
Rotations:
Graph the functions from the Bellringer with your graphing calculator and sketch below:
(1) f  x   x
(2) g  x    x
(3) h  x    x
(4) What is the effect of f(x)?
Sketch the following without a calculator:
(5) f(x) = –x2
(6) f  x   


x 3
Graph the following functions with your graphing calculator and sketch below:
(7) f(x) = 2x
(8) g(x) = 2x
(9) Compare #1 with #3 and #7 with #8. What is the effect of f(x)?
Sketch the following without a calculator:
(10) f(x) = (–x)2
(11) f  x    x  3
If the function h(x) has a domain [–4, 6] and range [–3, 10], what is the domain and range of
(12) –h(x)? D:
R:
Blackline Masters, Algebra II
(13) h(-x)? D:
R:
Page 7-8
Unit 7, Activity 3, Rotations Discovery Worksheet with Answers
Name
Date
Rotations:
Graph the functions from the Bellringer with your graphing calculator and sketch below:
(1) f  x   x
(2) g  x    x
(3) h  x    x
(4) What is the effect of f(x)?
Rotates the graph through space around the x-axis
Sketch the following without a calculator:
(6) f  x   
(5) f(x) = –x2


x 3
Graph the following functions with your graphing calculator and sketch below:
(7) f(x) = 2x
(8) g(x) = 2x
(9) Compare #1 with #3 and #7 with #8. What is the effect of f(x)? rotates graph around y-axis
Sketch the following without a calculator:
(10) f(x) = (–x)2
(11) f  x    x  3
If the function h(x) has a domain [–4, 6] and range [–3, 10], what is the domain and range of
(12) –h(x)? D: same
R:
Blackline Masters, Algebra II
[10, 3]
(13) h(x)? D:
[6, 4]
R: same
Page 7-9
Unit 7, Activity 3, Dilations Discovery Worksheet
Name
Date
Dilations:
Graph the following functions with your graphing calculator and sketch below:
(15) g  x   3 9  x2
(14) f  x   9  x 2
(17) s  x   9   3x 
1 
(18) j  x   9   x 
3 
2
(19) What is the effect of k f(x)
(16) h  x  
1
9  x2
3
2
if k > 1?
if 0 < k < 1?
(20) What is the effect of f(kx)
if k > 1?
if 0 < k < 1?
(21) Which one affects the domain?
range?
Sketch the following without your calculator for 6 < x < 6 and 4 < y < 4 and find (1½ , f (1½ )):
(22) t  x   x
(23) f  x   2 x
(24) g  x   2 x
If the function h(x) has a domain [–4, 6] and range [–3, 10], what is the domain and range of
1
(25) 5h(x)? D:
R:
(26) h  x  ? D:
R:
5
(27) h(5x)? D:
R:
Blackline Masters, Algebra II
1 
(28) h  x  ? D:
5 
R:
Page 7-10
Unit 7, Activity 3, Dilations Discovery Worksheet with Answers
Name
Date
Dilations:
Graph the following functions with your graphing calculator and sketch below:
(15) g  x   3 9  x2
(14) f  x   9  x 2
(17) s  x   9   3x 
2
1 
(18) j  x   9   x 
3 
.
(19) What is the effect of k f(x)
if k > 1?
(16) h  x  
1
9  x2
3
2
Graph #17 does touch the
xaxis, but it looks like it does
not because there are no pixels
near the zeroes
stretches the graph vertically
if 0 < k < 1? compresses the graph vertically
(20) What is the effect of f(kx)
if k > 1?
compresses the graph horizontally
if 0 < k < 1? stretches the graph horizontally
(21) Which one affects the domain? f(kx)
range? k f(x)
Sketch the following without your calculator for 6 < x < 6 and 4 < y < 4 and find (1½, f (1½ )):
(22) t  x   x
(23) f  x   2 x
(24) g  x   2 x
If the function h(x) has a domain [–4, 6] and range [–3, 10], what is the domain and range of
1
 3 
(25) 5h(x)? D: same
R: [15, 50]
(26) h  x  ? D: same
R:   , 2 
5
 5 
 4 6
1 
(27) h(5x)? D:   , 
R: same
(28) h  x  ? D: [20, 30] R: same
 5 5
5 
Blackline Masters, Algebra II
Page 7-11
Unit 7, Activity 3, Abstract Rotations & Dilations
Name
Date
Abstract Rotations & Dilations:
Domain of g(x):
Range of g(x):
Draw the graph of the following
over the graph of g(x), label the new points, and find the new domain and range:
(4, 8)
(4, 8)
g(x)
4
g(x)
4
(1, 2)
(1, 2)
(–5, –3)
(–5, –3)
(29) g(x) D:
(30) g(x) D:
R:
R:
(4, 8)
(4, 8)
g(x)
4
(1, 2)
(1, 2)
(–5, –3)
(–5, –3)
(31) 2g(x)
g(x)
4
D:
R:
(32) ½ g(x) D:
R:
(4, 8)
(4, 8)
g(x)
4
(1, 2)
(–5, –3)
(33) g(2x)
g(x)
4
(1, 2)
(–5, –3)
D:
R:
Blackline Masters, Algebra II
(34) g (½ x) D:
R:
Page 7-12
Unit 7, Activity 3, Abstract Rotations & Dilations with Answers
Name
Date
Abstract Rotations and Dilations:
Domain of g(x): [5, 4] Range of g(x): [3, 8]
Draw the graph of the following
over the graph of g(x), label the new points, and find the new domain and range:
(4, 8)
(4, 8)
(4, 8)
g(x)
4
(1, 2)
(1, 2)
(–5, –3)
(5, 3)
(29) g(x) D:
[4, 5]
g(x)
4
R:
same
(5, 3)
(1, 2)
(–5, –3)
(1, 2)
(30) g(x) D: same
[8, 3]
R:
(4, 8)
(4, 8)
8
4
(4, 8)
(1, 4) g(x)
g(x)
(4, 4)
4
(1, 2)
2
(–5, –1.5)
(–5, –3)
(–5, –3)
(1, 2)
(1, 1)
(–5, –6)
(31) 2g(x)
D: same
(2, 8)
4
R:
[6, 16]
(32) ½ g(x) D: same
R:
[1.5, 4]
(4, 8)
(4, 8)
g(x)
(1, 2)
(½ , 2)
(1, 2)
(2, 2)
(–10, –3) (–5, –3)
(–5, –3)
(–2.5, –3)
(33) g(2x)
D: [2.5, 2] R:
Blackline Masters, Algebra II
g(x)
4
same
(34) g (½ x) D: [10, 8]
R:
same
Page 7-13
(8, 8)
Unit 7, Activity 5, Tying It All Together
Name
Date
Reflections, Dilations, and Translations:
f(x) =
B
A
no change
I. Graphing: Given the graph of the
function f(x), match the following
shifts and translations.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
2f(x)
f(2x)
f(x)
f(x)
|f(x)|
f(|x|)
f(x) + 4
f(x + 4)
II. Domains and Ranges: Write the
new domain and range if g(x) has
a domain of [10, 4] and the
range is [6, 8]. If there is no
change, write “same.” If it cannot
be determined, write “CBD.”
DOMAIN:
1a)
b)
2a)
b)
3a)
b)
4a)
b)
5a)
b)
6a)
b)
C
D
E
F
G
H
I
J
K
L
RANGE:
g(x) + 1
g(x)  4
g(x + 1)
g(x  4)
g(2x)
2g(x)
g(½x)
½g(x)
-g(x)
g(x)
|g(x)|
g(|x|)
Blackline Masters, Algebra II
Page 7-14
Unit 7, Activity 5, Tying It All Together with Answers
Name
Date
Reflections, Dilations, and Translations:
f(x) =
B
A
no change
I. Graphing: Given the graph of the
function f(x), match the following
shifts and translations.
E
H
D
C
A
B
J
L
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
2f(x)
f(2x)
f(x)
f(x)
|f(x)|
f(|x|)
f(x) + 4
f(x + 4)
II. Domains and Ranges: Write the
new domain and range if g(x) has
a domain of [10, 4] and the
range is [6, 8]. If there is no
change, write “same.” If it cannot
be determined, write “CBD.”
1a)
b)
2a)
b)
3a)
b)
4a)
b)
5a)
b)
6a)
b)
g(x) + 1
g(x)  4
g(x + 1)
g(x  4)
g(2x)
2g(x)
g(½x)
½g(x)
g(x)
g(x)
|g(x)|
g(|x|)
DOMAIN:
same
same
[11, 3]
[6, 8]
[5, 2]
same
[20, 8]
same
same
[4, 10]
same
[4, 4]
RANGE:
[5, 9]
[10, 4]
same
same
same
[12, 16]
same
[3, 4]
[8, 6]
same
[0, 8]
CBD
Blackline Masters, Algebra II
C
D
E
F
G
H
I
J
K
L
Page 7-15
Unit 7, Activity 6, Picture the Pieces
Name
Date
Graphing Piecewise Functions:
In #1  4, graph and state the domain and range, x- and y-intercepts, the intervals on which the
function is increasing, decreasing, or constant, and if the function is continuous:
2 x 1
if x  1
 x  4 if x  4

(1) f  x    2
(2) f  x   
 x  8 if x  4
3 x  4  6 if x  1
(3)
 x3  5
if x  0

f  x   1
3 2 x  1 if 0  x  4

|  x  3  2 | if 0  x  5
f  x  
if x  0
 log( x  10)
2
(4)
(5) Graph h(x) and find the a and b that makes
the function continuous: a =
b=
 1
 x 1 x  2


h  x   ax  b 0  x  2


x0
 x

Blackline Masters, Algebra II
Page 7-16
Unit 7, Activity 6, Picture the Pieces
Analyzing Graphs of Piecewise Functions:
(6) Write a piecewise function for the graph of g(x) below.
(Assume all left endpoints are included and all right
endpoints are not included.)
 ___________ if

g  x    ___________ if
 ___________ if

_______
_______
_______
(7) Using the graph of g(x) above, draw the graph
h(x) = g(x+4) – 5 and write its piecewise function.
 ___________ if

h  x    ___________ if
 ___________ if

_______
_______
_______
(8) Using the graph of g(x) to the right, draw the
graph of t(x) = ½g(4x) and write its piecewise function.
 ___________ if

t  x    ___________ if
 ___________ if

_______
_______
_______
(9) Brett is on the ground outside the stadium and throws a
baseball to John at the top of the stadium 36 feet above the
ground. Brett throws with an initial velocity of 60 feet/sec.
It goes above John’s head, and he catches it on the way down.
John holds the ball for 5 seconds then drops it to Brett. Graph
the function and find a piecewise function that models the
height of the ball s(t) over time t in seconds after Brett throws
the ball. (Remember the quadratic equation from Unit 5 for
position of a free falling object if acceleration due to gravity
is –32 ft/sec2: s(t) = –16t2 + vot + so.)
(a) How long after Brett threw the ball did John catch it?
(b) How high did the ball go?
(c) At what time did the ball hit the ground?
 ___________________ if

s  t    ___________________ if
 ___________________ if

Blackline Masters, Algebra II
_______
_______
_______
Page 7-17
Unit 7, Activity 6, Picture the Pieces with Answers
Name
Date
Graphing Piecewise Functions:
In #1  4, graph and state the domain and range, x- and y-intercepts, the intervals on which the
function is increasing, decreasing, or constant, and if the function is continuous:
2 x 1
if x  1
 x  4 if x  4

(1) f  x    2
(2) f  x   
 x  8 if x  4
3 x  4  6 if x  1
D: all reals
R: [8, )
x-int: 2 2, 4
y-int: -8
inc:
(0,4)(4,∞)
dec: (-∞,0)
not continuous
(3)
 x3  5
if x  0

f  x   1
3 2 x  1 if 0  x  4

D: all reals
R: all reals
x-int: -6, -2
y-int: 2
inc: (-∞,-4)(-1,∞)
dec: (-4,-1)
not continuous
|  x  3  2 | if 0  x  5
f  x  
if x  0
 log( x  10)
2
(4)
D: (-∞, 4]
R: (-∞, 0){-1,2,5}
x-int: none
y-int: -1
inc: (-∞, 0)
constant: (0,2)(2,4)
not continuous
D: (-∞, 5]
R: (-∞, 7)
x-int: 9,3  2
y-int: log 10
inc: (-∞, 0)

3 
 
2,3  3  2,5
dec: 0,3  2,3  3,3  2
not continuous
(5) Graph h(x) and find the a and b that makes
the function continuous: a = ½
b=0
 1
 x 1


h  x   ax  b


 x

x2
0 x2
x0
Blackline Masters, Algebra II
 

Page 7-18

Unit 7, Activity 6, Picture the Pieces with Answers
Analyzing Graphs of Piecewise Functions:
(6) Write a piecewise function for the graph of g(x) below.
(Assume all left endpoints are included and all right
endpoints are not included.)
x  8
6  x  0

2

g  x    x  2 
0 x4

6.5
4 x9


(7) Using the graph of g(x) above, draw the graph
h(x) = g(x+4) – 5 and write its piecewise function.
x  7
10  x  4

2

h  x    x  2   5 4  x  0

1.5
0 x5


(8) Using the graph of g(x) to the right, draw the
graph of t(x) = ½g(4x) and write its piecewise function.

2 x  4

2
1
t  x     4x  2
2


3.25

1.5  x  0
0  x 1
1 x  9/ 4
(9) Brett is on the ground outside the stadium and throws a
baseball to John at the top of the stadium 36 feet above the
ground. Brett throws with an initial velocity of 60 feet/sec.
It goes above John’s head, and he catches it on the way down.
John holds the ball for 5 seconds then drops it to Brett. Graph
the function and find a piecewise function that models the
height of the ball s(t) over time t in seconds after Brett throws
the ball. (Remember the quadratic equation from Unit 5 for
position of a free falling object if acceleration due to gravity
is –32 ft/sec2: s(t) = –16t2 + vot + so.)
(a) How long after Brett threw the ball did John catch it?
3 seconds
(b) How high did the ball go?
56.250 feet
(c) At what time did the ball hit the ground? 9.500 sec
 16t 2  60t


s  t    36

2
16  t  8   36


Blackline Masters, Algebra II
0 x3
3 x 8
8  x  9.5
Page 7-19
Unit 7, Activity 7, Even & Odd Functions Discovery Worksheet
Name
Date
Rotations Revisited:
Graph the following in your notebook without a calculator:
y3 x
f(–x)
y = |–x|
y = (–x)3
y = 2 x
y  3 x
–f(x)
y = –|x|
y = –(x3)
y = (2x)
y  3 x
5
y x
y  x
y x
6
y
7
y= = x
1
x
y = (–x)
1
y   
x
y = –(x)
1
2
3
4
f(x)
y = |x|
y = x3
y = 2x
1
x
y
Even & Odd Functions Graphically:
Even Function ≡ any function in which f(–x) = f(x)
1. Look at the graphs of the functions above and in your bellringer, then list the parent
functions in which the graph of f(x) is the same as the graph of f(x) and are, therefore,
even functions.
2. Looking at the graphs of these even functions, they are symmetric to
.
Odd Function ≡ any function in which f(–x) = –f(x)
1. Look at the graphs of the functions above and in your bellringer, then list the parent
functions in which the graph of f(x) is the same as the graph of f(x) and are, therefore,
odd functions.
2. Looking at the graphs of these odd functions, they are symmetric to
.
3. Graph g(x) = x3 + 1 without a calculator and determine if it is even or odd; then explain
your answer.
4. Which of the parent functions are neither even nor odd?
Blackline Masters, Algebra II
Page 7-20
Unit 7, Activity 7, Even & Odd Functions Discovery Worksheet
Even & Odd Functions Numerically:
Consider the following table of values and determine which functions may be even, odd, or neither.
x
–3
–2
–1
0
1
2
3
f(x)
6
3
4
2
4
3
6
Even:
g(x)
6
–4
5
0
–5
4
–6
h(x)
6
5
4
3
2
1
6
Odd:
s(x)
6
5
–4
undefined
4
–5
–6
t(x)
–4
3
–2
5
–2
3
–4
Neither:
Even & Odd Functions Analytically:
Seven sets of ordered pairs are not sufficient to prove a function is even or odd. For example, in
h(x), h(–3) = h(3), but the rest did not work. In order to prove whether a function is even or odd,
substitute (–x) for every x and determine if f(–x) = f(x) or if f(–x) = –f(x) or neither. Analytically
determine if the following functions are even or odd, then graph on your calculator to check the
symmetry:
(1) f(x) = x4  3x2 + 5
(5) f(x) = x
(2)
f(x) = 4x3  x
(6)
f(x) = log |x|
(3)
f(x) = |x| + 5
(7)
f(x) = 3|x + 1|
(4)
f(x) = |x3|
(8)
f(x) = x3  4x2
Blackline Masters, Algebra II
Page 7-21
Unit 7, Activity 7, Even & Odd Functions Discovery Worksheet with Answers
Name
Date
Rotations Revisited:
Graph the following in your notebook without a calculator:
y3 x
f(–x)
y = |–x|
y = (–x)3
y = 2 x
y  3 x
–f(x)
y = –|x|
y = –(x3)
y = (2x)
y  3 x
5
y x
y  x
y x
6
y
7
y= = x
1
x
y = (–x)
1
y   
x
y = –(x)
1
2
3
4
f(x)
y = |x|
y = x3
y = 2x
1
x
y
Even & Odd Functions Graphically:
Even Function ≡ any function in which f(–x) = f(x)
1. Look at the graphs of the functions above and in your bellringer, then list the parent
functions in which the graph of f(x) is the same as the graph of f(x) and are, therefore,
even functions.
f(x) = x2 , f(x) = |x|
2. Looking at the graphs of these even functions, they are symmetric to
yaxis
.
Odd Function ≡ any function in which f(–x) = –f(x)
1. Look at the graphs of the functions above and in your bellringer, then list the parent
functions in which the graph of f(x) is the same as the graph of f(x) and are, therefore,
odd functions.
1
f(x) = x3, f  x   3 x , f  x   , f(x) = x
x
2. Looking at the graphs of these odd functions, they are symmetric to the origin
.
3. Graph g(x) = x3 + 1 without a calculator and determine if it is even or odd; then explain
your answer.
Neither even nor odd. Not symmetric to yaxis nor origin.
4. Which of the parent functions are neither even nor odd?
f(x) = log x, f(x) = 2x, f  x   x ,
Blackline Masters, Algebra II
Page 7-22
Unit 7, Activity 7, Even & Odd Functions Discovery Worksheet with Answers
Even & Odd Functions Numerically:
Consider the following table of values and determine which functions may be even, odd, or neither.
x
–3
–2
–1
0
1
2
3
Even:
f(x)
6
3
4
2
4
3
6
f(x) and t(x)
g(x)
6
–4
5
0
–5
4
–6
Odd:
h(x)
6
5
4
3
2
1
6
g(x) and s(x)
s(x)
6
5
–4
undefined
4
–5
–6
Neither:
t(x)
–4
3
–2
5
–2
3
–4
h(x)
Even & Odd Functions Analytically:
Seven sets of ordered pairs are not sufficient to prove a function is even or odd. For example, in
h(x), h(–3) = h(3), but the rest did not work. In order to prove whether a function is even or odd,
substitute (–x) for every x and determine if f(–x) = f(x) or if f(–x) = –f(x) or neither. Analytically
determine if the following functions are even or odd, then graph on your calculator to check the
symmetry:
(1) f(x) = x4  3x2 + 5
(5) f(x) = x
f(x) = (x)4  3(x)2 + 5
f  x 
x  | x |  f  x 
= x4  3x2 + 5 = f(x)
 even, symmetric to yaxis
 even, symmetric to yaxis
(2)
f(x) = x3  4x
f(x) = (x)3 + 4(x)
= (x3 + 4x) = f(x)
 odd, symmetric to the origin
(6) f(x) = log |x|
f(x) = log |x|
= log |x| = f(x)
 even, symmetric to yaxis
(3)
f(x) = |x| + 5
f(x) = |x| + 5
= |x| + 5 = f(x)
 even, symmetric to yaxis
(7) f(x) = 3|x + 1|
f(x) = 3|x + 1| = 3|(x  1|
= 3|x  1|≠ f(x) and ≠ f(x)
 neither even nor odd
(4) f(x) = |x3|
f(x) = |(x)3|
= |-x3| = |x3|= f(x)
 even, symmetric to yaxis
Blackline Masters, Algebra II
(8) f(x) = x3  4x2
f(x) = (x)3  4(x)2
= x3  4x2 ≠ f(x) and ≠ f(x)
 neither even nor odd
Page 7-23
Unit 7, Activity 8, Modeling to Predict the Future
Data Analysis Research Project:
Objectives:
1. Collect data for the past twenty years concerning a topic selected from the list below.
2. Create a mathematical model.
3. Trace the history of the statistics.
4. Evaluate the future impact.
5. Create a PowerPoint® presentation of the data including pictures, history, economic
impact, spreadsheet data, regression graph and equation, and future predictions.
Possible Topics:
1. # of deaths by carbon monoxide poisoning (overall, in the house, in a car, in a boat)
2. boating accidents or deaths (in LA or in US)
3. jet ski accidents or deaths (in LA or in US)
4. drownings (in LA or in US)
5. number of registered boats (in LA or in US)
6. drunken driving (accidents or deaths, in this parish , LA or in US)
7. DWIs (in this parish, LA or in US)
8. car accidents or deaths (in this town, this parish, LA or in US)
9. suicides (in this town, this parish, LA or in US)
10. census statistics such as population, population by race, marriages, divorces, births,
deaths, lifespan, (in this town, this parish, LA or in US)
11. electricity usage (in this town, this parish, LA or in US)
12. land value (in this town, this parish, LA or in US)
13. animal population (in this town, this parish, LA or in US)
14. deaths by any other cause (choose the disease or cause, in this parish, LA or in US)
15. obesity
16. drop out rates
17. building permits for new houses (choose in this town, this parish, LA or in US)
Research:
1. This is an individual or pair project, and each person/pair must have different data.
2. Research on the Internet or other resource to find at least twenty data points; the more
data you have, the better the mathematical model will be. The youngest data should be no
more than five years ago.
3. Research the historical significance of the data and determine why it would be increasing
and decreasing at different times, etc. Determine what might have been happening
historically in a year when there is an obvious outlier.
4. Take pictures with a digital camera or find pictures on the Internet to use on your
PowerPoint®
Calculator/Computer Data Analysis:
1. Enter the data into a spreadsheet or your calculator. Time should be your independent
variable using 1 for 1991, 2 for 1992, etc.
2. Create a scatter point chart, find the mathematical model for the data (regression equation
or trendline), and find the correlation coefficient (Rsquared value on spreadsheet).
Blackline Masters, Algebra II
Page 7-24
Unit 7, Activity 8, Modeling to Predict the Future
3. Use a model that has the characteristics you want such as increasing or decreasing,
correct endbehavior, zeros, etc. (For a better regression equation, you may have to
eliminate outliers, create two regression equations, one with and one without the outlier,
and compare, or create a piecewise function.)
Extrapolation:
1. Use your mathematical model to predict what will happen if the trend continues for the
next five years and explain the feasibility and limitations of the predictions.
2. Discuss what outliers may occur that would affect this extrapolation.
Presentation:
1. Create a six slide PowerPoint® presentation. Make sure to use colors that show well
when projected on the screen, and use a large font size.
Slide 1: Introduction of the topic with a relative picture (not clip art), your name, date,
class, hour.
Slide 2: Statement of the problem, history, and economics of the topic (Use bullets,
not sentences, to help you in your oral report – no more than 15 words, bullets
should enter PowerPoint® one at a time as you talk.)
Slide 3: Scatter plot graph of the data-clearly labeled, curve, regression equation, and
correlation coefficient. Type regression equation with 3 decimal places on the
slide not on the chart. (Use proper scientific notation if necessary, no E’s, in
the equation.) Be able to discuss why you chose this function to model your
data based on its characteristics.
Slide 4: Prediction for five years from now if the trend continues. (Show your equation
with independent variable plugged in.) Discuss reasonableness.
Slide 56: Any other pertinent info, your data, URL for links to other sites for additional
information, or another data comparison. Include resources used to find data
and how your data could help with solutions to particular problems.
2. Present your project to your Algebra II class and to another class. You may not read from
the PowerPoint® or from a paper – use index cards to help you present. Dress nicely on
presentation day.
Project Analysis: Type a discussion concerning what you learned mathematically, historically,
and technologically, and express your opinion of how to improve the project.
Timeline:
1. Three days from now, bring copy data to class along with a problem statement (why you
are examining this data), so it can be approved, and you can begin working on it in class.
You will hand this in so make a copy.
2. Project is due on _____________________.
Final Product:
1. Disk or flash drive containing the PowerPoint® presentation or email it to me – it should be
saved as “your name/s and title of presentation.”
2. A printout of slides in the presentation. (the “Handout” printout, not a page for each slide).
3. Release forms signed by all people in the photographs.
4. Project Analysis
5. Rubric
Blackline Masters, Algebra II
Page 7-25
Unit 7, Activity 8, Modeling to Predict the Future Rubric
Name
Date
Evaluation Rubric:
Written work to be handed in
Data and Problem statement explaining why you are examining this
data handed in three days after assigned
Project Analysis concerning what you learned mathematically,
historically, and technologically, and expressing your opinion of how
to improve the project
Printout of PowerPoint® presentation, use of easytoread colors and
fonts and release forms signed by all people in digital pictures
PowerPoint®
Slide 1: Introduction of the topic with a relative picture (not clip art),
your name, date, class, hour
Slide 2: Statement of the problem, history, and economics of the topic,
bullets, not sentences, entering one at a time, no more than 15 words
Slide 3: Scatter plot graph of the data clearly labeled, curve, regression
equation, and correlation coefficient. Type regression equation with 3
decimal places on the slide not on the chart. (Use proper scientific
notation if necessary, no E’s, in the equation.)
Slide 4: Prediction for five years from now if the trend continues.
(Show your equation with independent variable plugged in.) Discuss
reasonableness.
Slide 56: Any other pertinent info, your data, URL for links to other
sites for additional information, or another data comparison. Include
resources used to find data. How your data could help with solutions to
particular problems
Presentation
Verbal presentation accompanying PowerPoint® (concise, complete,
acceptable language, not read from PowerPoint® or paper, dressed
nicely)
Presentation to another class
Handed in on time
TOTAL
Blackline Masters, Algebra II
Teacher
Rating
xxxxxx
Possible
Points
xxxxxx
10
10
10
xxxxxx
xxxxxx
10
10
10
10
10
xxxxxx
xxxxxx
10
10
(-10/day
late)
100
Page 7-26
Unit 7, Activity 9, Discovering Trigonometric Graphs
Name
Date
Vocabulary Self-Awareness Chart
Complete the following chart with a partner.
 Rate your understanding of each concept with either a “+” (understand well), “”
(limited understanding or unsure), or a “” (don’t know)
 Write a short description of each term.
Mathematical Terms
1
modeling
2
scatter plot
3
best fit equation
4
regression equation
5
interpolate
6
extrapolate
7
translation
8
reflection
9
dilation
10
periodic function
11
fundamental period
12
frequency
13
amplitude
14
midline
15
sin 
16
cos 
sinusoidal curve
17
+
 
Short description in your own words
New Characteristics of a Graph
(1) Is this graph periodic?
(2) What is the period?
(3) What is the frequency?
(4) Where is the midline?
(5) What is the amplitude?
Blackline Masters, Algebra II
Page 7-27
Unit 7, Activity 9, Discovering Trigonometric Graphs
Trigonometric Graphs
Graph the following functions on your graphing calculator:
 set the mode to radians and ZOOM Trig
 sketch the graph below identifying the x- intercepts as multiples of 
 identify the y-intercepts
 locate the ordered pairs that are relative maximum and minimum points
 answer the questions below.
f(x) = sin x
f(x) = cos x
(1) Are the graphs periodic?
(2) What is the period?
(3) What is the frequency?
(4) Where is the midline?
(5) What is the amplitude?
Blackline Masters, Algebra II
Page 7-28
Unit 7, Activity 9, Discovering Trigonometric Graphs with Answers
Name
Date
Vocabulary Self-Awareness Chart
Complete the following chart with a partner.
 Rate your understanding of each concept with either a “+” (understand well), “”
(limited understanding or unsure), or a “” (don’t know)
 Write a short description of each term.
Mathematical Terms
1
modeling
2
scatter plot
3
best fit equation
4
regression equation
5
interpolate
6
extrapolate
7
translation
8
reflection
9
dilation
10
periodic
11
frequency
12
amplitude
13
midline
14
sin 
15
cos 
sinusoidal curve
16
 
+
Short description in your own words
answers will vary based on student’s wording
New Characteristics of a Graph
yes
(1) Is this graph periodic?
(2) What is the period?
8 units
(3) What is the frequency?
1/8
(4) Where is the midline?
y=2
(5) What is the amplitude?
6 units
Blackline Masters, Algebra II
Page 7-29
Unit 7, Activity 9, Discovering Trigonometric Graphs with Answers
Trigonometric Graphs
Graph the following functions on your graphing calculator:
 set the mode to radians and ZOOM Trig
 sketch the graph below identifying the x- intercepts as multiples of 
 identify the y-intercepts
 locate the ordered pairs that are relative maximum and minimum points
 answer the questions that follow.
(1) Are the graphs periodic?
yes
(2) What is the period?
2
(3) What is the frequency?
1/(2)
(4) Where is the midline?
y=0
(5) What is the amplitude?
1
Blackline Masters, Algebra II
Page 7-30
Unit 7, Activity 9, Modeling with Trigonometric Functions
Name
Date
Translations and Dilations of Trig Functions
State what happens to a graph in the following situations (k>0):
(1)
f(x) + k
(2)
f(x - k)
(3)
f (kx)
(4)
k f(x)
Using the above information, sketch the following translations and dilations on the parent graphs
below, then check your answer on your graphing calculator:
(5) f(x) = (sin x) + 2
Where is the new midline? y = _______
(6) f(x) = sin (x - )
(7) f(x) = 2 sin x
What is the new amplitude? _____
(8) f(x) = sin 2x
What is the new period? ______ What is the new frequency? _____
(9) Considering the general trig function f(x) = A sin B(x – C) + D or f(x) = A cos B(x – C) + D,
which constant affects the
amplitude? _____ period? 2/_____ horizontal shift? _____ vertical shift (midline)? _____
Blackline Masters, Algebra II
Page 7-31
Unit 7, Activity 9, Modeling with Trigonometric Functions
Real-World Models Using Trig Functions
depth of water in ft.
High and Low Tides: The depth of the water at the end of the wharf varies with the tides
throughout the day. Today the high tide occurs at 4:00 a.m. with a depth of 15 ft. The low tide
occurs at 10:00 a.m. with a depth of 7.0 ft.
(a) Sketch one period of the graph that models the depth d(t) of the water t hours after midnight.
hours after midnight
(b) What is the amplitude? _____ period? _____ horizontal shift? ________ midline? _______
(c) Write an equation for your curve. d(t) =
(d) Find the depth of the water at noon:
cos
(t -
)+
(Use your calculator in radian mode.)
ht. above ground in ft.
Ferris Wheel Problem: Picture yourself on a Ferris wheel similar to the picture. When the last
seat is filled, your seat is somewhere on the right side of the wheel. The Ferris wheel starts going
counterclockwise, and you find that it takes you 3 seconds to reach the top and the wheel makes
a revolution every 8 seconds.
(a) Sketch one period of the graph of your distance from the ground over time.
50 ft
10 ft
sec. after ride starts
(b) What is the amplitude? _____ period? _____ horizontal shift? _____ midline?_____
(c) Write an equation for your curve. d(t) = ___cos___(t – ___) + __
_
(d) Predict your height above ground when t = 6 seconds:
Blackline Masters, Algebra II
Page 7-32
Unit 7, Activity 9, Modeling with Trigonometric Functions with Answers
Name
Date
Translations and Dilations of Trig Functions
State what happens to a graph in the following situations (k>0):
(1)
f(x) + k
vertical shift up k
(2)
f(x - k)
horizontal shift right k
(3)
f (kx)
shrink horizontally
(4)
k f(x)
stretch vertically
Using the above information, sketch the following translations and dilations on the parent graphs
below then check your answer on your graphing calculator:
(9) Considering the general trig function f(x) = A sin B(x – C) + D or f(x)=A cos B(x – C) + D,
which constant affects the
amplitude? __A_ period? 2/_B__ horizontal shift? C right vertical shift (midline)? up D
Blackline Masters, Algebra II
Page 7-33
Unit 7, Activity 9, Modeling with Trigonometric Functions with Answers
Real-World Models Using Trig Functions
High and Low Tides: The depth of the water at the end of the wharf varies with the tides
throughout the day. Today the high tide occurs at 4:00 a.m. with a depth of 15 ft. The low tide
occurs at 10:00 a.m. with a depth of 7.0 ft.
(a) Sketch one period of the graph that models the depth d(t) of the water t hours after midnight.
(b) What is the amplitude? 4 ft_ period? _12 hrs. horizontal shift? right 4 hrs_ midline? y=11 ft.
(c) Write an equation for your curve. d  t   4 cos
(d) Find the depth of the water at noon:
9 ft.
2
 t  4  11
12
(Use your calculator in radian mode.)
Ferris Wheel Problem: Picture yourself on a Ferris wheel similar to the picture. When the last
seat is filled, your seat is somewhere on the right side of the wheel. The Ferris wheel starts going
counterclockwise, and you find that it takes you 5 seconds to reach the top and the wheel makes
a revolution every 35 seconds.
(a) Sketch one period of the graph of your distance from the ground over time.
50 ft
10 ft
(b) What is the amplitude? 20 ft period? 35 sec. horizontal shift? right 5 sec. midline? y = 30 ft.
2
(c) Write an equation for your curve. d  t   20 cos
 t  5  30
35
d) Predict your height above ground when t = 15 seconds:
25.5 ft.
Blackline Masters, Algebra II
Page 7-34
Unit 7, Activity 10, Pythagorean Identity for Trig Functions
Name
Date
Pythagorean Theorem:
Use the right triangle to the right to answer the following questions:

(1) Pythagorean Theorem using x and y:
(2) Trig ratios using x and y: cos   ______ sin  ______ tan  _____
So you can conclude on a circle of radius 1, cos  is the x-coordinate and sin  is the y-coordinate.
(3) Pythagorean Identity using cos  and sin  :
Does the Pythagorean Identity hold even if the circle has a radius  1?
(4) Pythagorean Theorem using x and y:
(5) Trig ratios using x and y: tan  ______
cos   ______  x = ________ sin  ______  y = ______
(6) Pythagorean Identity using cos  and sin  :
4

cos , sin , and tan  in any Quadrant
In geometry you think of x and y‘s as lengths, so they would be sides of the
triangle in the first quadrant. However, if you use the idea that on any circle of
radius 1 and angle  formed by the x-axis and the hypotenuse, you can define
y
x as cos  and y as sin  and
as tan . Therefore, sin , cos , and tan  can
x
be positive or negative based on the quadrant in which you draw the right
triangle. Complete the following chart:
Signs of Function Values in Quadrants
Trigonometric
Function
cos 
sin 
tan 
Quadrant I
Blackline Masters, Algebra II
Quadrant II
Quadrant III
Quadrant IV
Page 7-35
Unit 7, Activity 10, Pythagorean Identity for Trig Functions with Answers
Name
Date
Pythagorean Theorem:
Use the right triangle to the right to answer the following questions:

x2 + y2 = 1
(1) Pythagorean Theorem using x and y:
x
y
y
sin 
tan 
1
1
x
So you can conclude on a circle of radius 1, cos  is the x-coordinate and sin  is the y-coordinate.
(2) Trig ratios using x and y: cos  
(3) Pythagorean Identity using cos  and sin  :  cos     sin    1
2
2
Does the Pythagorean Identity hold even if the circle has a radius  1?
(4) Pythagorean Theorem using x and y:
x2 + y2 = 16
(5) Trig ratios using x and y: tan 
y
x
4
cos   x  x = 4 cos  sin  y  y = 4 sin 
4

4
(6) Pythagorean Identity using cos  and sin  :
 4 cos  
2
  4 sin    16   cos     sin    1
2
2
2
cos , sin , and tan  in any Quadrant
In geometry you think of x and y as lengths so they would be sides of the
triangle in the first quadrant. However, if you use the idea that on any circle of
radius 1 and angle  formed by the x-axis and the hypotenuse, you can define
x as cos  and y as sin  and y as tan . Therefore, sin , cos , and tan  can
x
be positive or negative based on the quadrant in which you draw the right
triangle. Complete the following chart:
Signs of Function Values in Quadrants
Trigonometric
Function
cos 
sin 
tan 
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
+
+
+
+
-
+
+
-
Blackline Masters, Algebra II
Page 7-36
Unit 7, Activity 10, Properties of Functions
Name
Date
Properties of Functions Word Grid
Place an “X” in the box corresponding to the property illustrated by the function:
domain
all reals
range
all reals
increasing
decreasing
odd
even
same
end
behavior
opposite
end
behavior
periodic
f(x) = x
f(x) = x2
f ( x)  x
f(x) = x3
f(x) = |x|
f(x) = 2x
f ( x) 
1
x
f  x  3 x
f(x) = log x
f ( x)  x
f(x) = sin x
f(x) = cos x
Blackline Masters, Algebra II
Page 7-37
Unit 7, Activity 10, Properties of Functions with Answers
Name
Date
Properties of Functions Word Grid
Place an “X” in the box corresponding to the property illustrated by the function:
domain
all reals
range
all reals
increasing
f(x) = x
X
X
X
f(x) = x2
X
even
same
end
behavior
X
opposite
end
behavior
periodic
X
X
X
f(x) = x3
X
f(x) = |x|
X
f(x) = 2x
X
X
X
X
X
f(x) = log x
f ( x)  x
X
f(x) = sin x
X
f(x) = cos x
X
X
X
X
X
Blackline Masters, Algebra II
X
X
X
X
X
X
1
x
f  x  3 x
odd
X
f ( x)  x
f ( x) 
decreasing
X
X
X
X
X
X
X
X
X
Page 7-38