Corrosion Engineering Principles and Practice McGrawHill 2008 0071482431
McGrawHill Grade 11 Functions Unit 3
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Transcript of McGrawHill Grade 11 Functions Unit 3
147
graph an exponential relation, given its equation in the form y = ax (a > 0, a ≠ 1), define this relation as the function f (x) = ax, and explain why it is a function
determine the value of a power with a rational exponent
simplify algebraic expressions containing integer and rational exponents and evaluate numerical expressions containing integer and rational exponents and rational bases
determine and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes for exponential functions represented in a variety of ways
distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways
determine and describe the roles of the parameters a, k, d, and c in functions of the form y = af [k(x — d )] + c in terms of transformations on the graph of f (x) = ax (a > 0, a ≠ 1)
sketch graphs of y = af [k(x — d )] + c by applying one or more transformations to the graph of f (x) = ax (a > 0, a ≠ 1), and state the domain and range of the transformed functions
determine that the equation of a given exponential function can be expressed using different bases
represent an exponential function with an equation, given its graph or its properties
collect data that can be modelled as an exponential function, from primary sources, using a variety of tools, or from secondary sources, and graph the data
identify exponential functions, including those that arise from real-world applications involving growth and decay, given various representations, and explain any restrictions that the context places on the domain and range
solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications by interpreting the graphs or by substituting values for the exponent into the equations
Exponential
Functions
CHAPTER
3
By the end of this chapter, you will
Refer to the Prerequisite Skills Appendix on pages 478 to 495 for examples of the topics and further practice.
Exponent Rules
1.
Rulea) xa xb
b) xa
xb
c) xa b
Method
A xa b
B xa b
C xa b
2.
3.
a) x x b) y y y
c) m m d) hh
e) a a b b f) x yx y
g) ab c h) uv
i) ab j) w
r
4.
a) b)
c) d)
e) f)
g) h)
Zero and Negative Exponents
5. y x
a)
x y
4 24 = 16
3 23 = 8
2 22 =
1 21 =
0 20 =
b) y
c)
x y
4 24 = 16
3 23 = 8
2 22 =
1 21 =
0 20 =
—1 2—1 = 1
2 = 1
21
—2 2—2 = 1
4 =
—3 2—3 =
—n 2—n =
6.
a) b)
c) d)
e) f)
g) h)
7.
a) x x b) y
c) u v u v
d) a b
148 MHR • Functions 11 • Chapter 3
Prerequisite Skills
Graph Functions
8.
i)
ii)
iii) x y
a)
2
y
x42 860—4 —2—8 —6
4
b)
2
—4
y
x42 860—4 —2—6
4
—2
—8
9.
i)
ii)
iii) x y
a) y x b) y x
Transformations of Functions
10. y x
2
—4
—2
x420—4 —2
4
y
11. y x
a) y x b) y x
c) y x d) y x
e) y x f) y x
Chapter Problem
Prerequisite Skills • MHR 149
Investigate
How can you discover the nature of exponential growth?
Pattern 3
Pattern 1 Pattern 2
1.
2.
a)
b)
3. a)
b)
The Nature of Exponential Growth
150 MHR • Functions 11 • Chapter 3
3.1
Tools
• coloured tiles or linking cubes
• grid paper
or
• graphing calculator
or
• computer with The Geometer’s Sketchpad®
4. a)
Pattern 1
Term number,
n
Number of Squares,
t
1 2
2 4
3 6
4
5
b)
c)
d) tn
e)
5.
6. Reflect
a)
b)
c)
Example 1
Model Exponential Growth
a)
b)
c)
d)
i)
ii)
e)
3.1 The Nature of Exponential Growth • MHR 151
First Differences
2
2
Second Differences
0
152 MHR • Functions 11 • Chapter 3
Solution
a)
Day Population
0 100
1 100 3 = 300
2 300 3 = 900
3 900 3 = 2700
4 2700 3 = 8100
b)
Day Population
0 100
1 100 31 = 300
2 100 32 = 900
3 100 33 = 2700
4 100 34 = 8100
n 100 3n
pn p n n
c) p n n
d) n
p n n
i) n
p
ii) n
p
The initial population is 100.
The population triples each day.
After 1 day, the initial population triples.
After 2 days, the initial population triples again.
After n days, the initial population has tripled n times.
5400
7200
p
3600
1800
n20 4
9000
p(n) = 100 3n
e)
Day Population
0 100
1 300
2 900
3 2700
4 8100
exponential growth
Example 2
Apply a Zero Exponent in a Model
Solution
Day Population
4 100 34 = 8100
3 100 33 = 2700
2 100 32 = 900
1 100 31 = 300
0 100 30 = 100
y x y
x y
Technology TipYou can use a graphing calculator to calculate finite differences.
Refer to the Technology Appendix on pages 496 to 516.
exponential growth pattern of growth in
which each term is multiplied by a constant amount (greater than one) to produce the next term
produces a graph that increases at a constantly increasing rate
has a repeating pattern of finite differences: the ratio of successive finite differences is constant
y
x0
Technology TipYou can use a graphing calculator to view and analyse this function. To find the y-intercept:
Press 2nd [CALC].
Select 1:value and enter 0 to identify the y-intercept.
For this pattern to be extended, 30 must equal 1, because 100 1 = 100.
3.1 The Nature of Exponential Growth • MHR 153
First Differences
300 — 100 = 200
900 — 300 = 600
2700 — 900 = 1800
8100 — 2700 = 5400
Second Differences
600 — 200 = 400
1800 — 600 = 1200
5400 — 1800 = 3600
154 MHR • Functions 11 • Chapter 3
Key Concepts
b b b
y
x0
Communicate Your Understanding
C1
a)
A p n n
B p n n
C p n n
b)
C2
y x y x y x
a)
b)
C3
y x y x y x
C4
a)
b)
c)
C5
A Practise
For help with question 1, refer to Example 1.
1.
a)
Day Population
0 20
1 80
2
3
4
5
b)
c)
d)
e)
For help with questions 2 to 4, refer to Example 2.
2.
3. a) aa
b)
c) aa
d)
4.
a) b) c) d) x
B Connect and Apply 5. a)
t n n nt n
b) t
c)
i) ii)
d)
e)
6. Use Technology
a)y x
i)
ii)
iii)
b)
c) y x
3.1 The Nature of Exponential Growth • MHR 155
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
First
Differences Second
Differences
156 MHR • Functions 11 • Chapter 3
7.
a)
i) ii)
b)
c)
8.
9.
a)
p t
b)
c)
d)
10.
A
A P i n Pi
n
a)
b)
Number of
Compounding
Periods
(years) Amount ($)
0 100
1 105
2
3
4
c)
d)
e)
f)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
A = 100(1 + 0.05)1
= 100(1.05)1
= 105
11.
a) b)
12.
a)
b)
I Prt IP r
t
C Extend 13.
14.
a)
b)
15.
a)
b)
16. Math Contest
A B C D
17. Math Contest
A B C D
18. Math Contestx y
z
x zy
x y z
A B C D
19. Math Contest
20. Math Contest
n n
3.1 The Nature of Exponential Growth • MHR 157
Use Technology
158 MHR • Functions 11 • Chapter 3
Use the Lists and Trace Features on a TI-NspireTM CAS Graphing
Calculator
A: Calculate First and Second Differences for the Population Data in Example 1 on pages 151 to 153
1. Lists & Spreadsheet
2.
day ·
pop ·
3.
first_diff
k
List ·
Operations·
Difference List·
pop·
Tools
• TI-NspireTM CAS graphing calculator
B: Calculate the Population Data in Example 2 on page 153
1. Lists & Spreadsheet
2.
day
pop
·
C: Trace the Population Function in Example 2
1. Graphs & Geometry
2.
f1 x ·
b 4:Window 1:Window Settings
x y
b 5:Trace 1:Graph Trace
Use Technology: Use the Lists and Trace Features on a TI-Nspire™ CAS Graphing Calculator • MHR 159
160 MHR • Functions 11 • Chapter 3
Exponential Decay: Connecting to Negative Exponents
Investigate
How can you explore exponential decay?
half-life
1.
Time
(years)
Number of Half-Life Periods
(2 years)
Amount of U-239 Remaining
(mg)
0 0 1000
2 1 500
4 2
6 3
8 4
2. a)
b)
•
3.2
Tools
• graphing calculator
half-life length of time for an
unstable element to spontaneously decay to one half its original amount
Connections
You will learn more about nuclear fission if you study physics in high school or university.Visit the Functions 11 page on the McGraw-Hill Ryerson Web site and follow the links to Chapter 3 to find out more about CANDU reactors.
3. a)
i) ii)
b)
4. a)
• STAT EDIT
• L1 L2
• 2nd Plot1
•
• ZOOM 9:ZoomStat
b) Reflect
5. a)
b)
6. a)
• STAT
CALC
• 0:ExpReg
• 2nd , 2nd , VARS
Y-VARS 1:Function
• 1:Y1 ENTER
b) a by abx
c) Reflect a b
7. a) GRAPH
b)
8. Reflect Zoom Trace
3.2 Exponential Decay: Connecting to Negative Exponents • MHR 161
Technology Tip
Press 2nd MODE for [QUIT] to return to the home screen from other screens.
162 MHR • Functions 11 • Chapter 3
exponential decay
Example 1
Model Exponential Decay
Solution
Method 1: Use a Table or a Spreadsheet
Enter
exponential decay a pattern of decay in
which each term is multiplied by a constant fraction between zero and one to produce the next term
produces a graph that decreases at a constantly decreasing rate
has a repeating exponential pattern of finite differences: the ratio of successive finite differences is constant
y
x0
3.2 Exponential Decay: Connecting to Negative Exponents • MHR 163
Method 2: Use Systematic Trial
A nn
n A
A n
A nn
n
n
n
n 0.5n Mathematical Reasoning
5 0.03125 Try 5 half-lives. This gives a value much lower than 0.1. Try a
shorter period of time.
2 0.25 Too high. The correct value is between 2 and 5. Try 3 next.
3 0.125 Close, but high.
3.3 0.1015… This is very close to 0.1. This will give a reasonable approximation.
Method 3: Use a Graphical Model
A nn
Trace
Divide both sides by 50.
Connections
You will learn how to find exact solutions to equations such as
5 = 50 1
2 n
when you
study logarithms in grade 12.
164 MHR • Functions 11 • Chapter 3
A nn
n
b n
bn b b n
Example 2
Evaluate Expressions Involving Negative Exponents
a) b) c) d)
Solution
a)
b)
c)
d)
Apply the product rule. Add the exponents.
Apply the power of a power rule first. Multiply exponents.
Apply the quotient rule. Subtract exponents.
Example 3
Simplify Expressions Involving Negative Exponents
a) x x x b)a ba b
c) u v
Solution
a) x x x x x –
x
b) a ba b
a b
a b
ab
c) u v u v
u v
vu
Example 4
Evaluate Powers Involving Fractional Bases
a) b)
Solution
a)
b)
Apply the product rule.
Apply the quotient rule.
Apply the power of a power rule.
Write with a positive exponent.
Multiply by the reciprocal.
Apply the property of negative exponents.
3.2 Exponential Decay: Connecting to Negative Exponents • MHR 165
Connections
The result in Example 4 part a) can be generalized as follows.
a b
n
b a
n
for
a, b , a, b 0, and n .
166 MHR • Functions 11 • Chapter 3
Key Concepts
b n
bn b b n
a b
n b a n
a b a b n
y
x0
Communicate Your Understanding
C1
C2 A nn
A
n
A n n
a)
b)
C3
C4
A Practise
For help with questions 1 to 4, refer to Example 2.
1.
a) b) x c) y
d) ab e) x f) x
2.
a) b) k
3.
a) b) c)
d) e) f)
4.
a) b)
c) d)
e) f)
g) h)
For help with question 5, refer to Example 3.
5.
a) m m b) v v
c) p p d) ww
e) k f) ab
For help with questions 6 and 7, refer to Example 4.
6.
a) b) c)
d) e) f)
7.
a)ab
b) u
c)gw
d) ab
e) ab
f) x y
B Connect and Apply 8.
a)
b) f x abx
c)
d)
e)
f)
9. a)
i) ii)
b)
c)
d)
e)
10.
11.
3.2 Exponential Decay: Connecting to Negative Exponents • MHR 167
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
168 MHR • Functions 11 • Chapter 3
12.
v tv t t
a)
b)
i) ii)
c)
d)
13.
a)
b)
i)
ii)
14.
a b
n b a n
a b
a b n
15. A P i n
P A i n
Achievement Check
16. P n A i n
P n
A
in
a)
i)
ii)
b)
c) A
17.
18. Chapter Problem
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
F GMmr
GM m
r
Mm
F
C Extend
19.
a) F GMmr
b)
c)
20.
a)
b) r
21. Math Contest f x x
f x f x
A B C D
22. Math Contest y
yx
A B C D
23. Math Contest
x xx xx xx xx x x
x x
A B C D
3.2 Exponential Decay: Connecting to Negative Exponents • MHR 169
Career Connection
170 MHR • Functions 11 • Chapter 3
3.3
Rational Exponents
Investigate
How can you find the meaning of a rational exponent?
1.
2.
a) P
b)
i)
ii)
c)
3. a)
b) t P t
1200
1600
2000
P
800
400
t20 4
P(t) = 100(4)t
Technology TipIf you use a graphing calculator, you can use the Zoom and Trace operations to find the coordinates of points with greater accuracy.
4. a)
b)
c) Reflect
5. a)
b) Reflect
6. a) Reflect
b)
c)
b b b b
Example 1
Powers With Rational Exponents of the Form 1
n
a) b) c)
d) e)
Solution
a)
3.3 Rational Exponents • MHR 171
The cube of 8 1
3 is equal to 8, so 8 1
3 is equal to the cube root of 8.
Connections
3
__
8 is read as “the cube root of eight.”
4
___
16 is read as “the fourth root of sixteen.”
The terminology used for this type of expression is as follows:
4
___
16 is called a radical. 16 is the radicand. 4 is the index.
172 MHR • Functions 11 • Chapter 3
b)
c)
d)
e)
Example 2
Powers With Rational Exponents of the Form m
n
a) b) c)
Solution
a)
b)
Think: What number raised to the exponent 5 produces —32?
5
____
—32 = —2, because (—2)(—2)(—2)(—2)(—2) = —32.
4
____
16
81 = 2
3 , because 2
3 4
= 2
3 2
3 2
3 2
3
= 2 2 2 2
3 3 3 3
= 16
81
Rewrite the power with a positive exponent.
Evaluate the cube root in the denominator.
Express the exponent as a product: 2
3 = 1
3 2
Use the power of a power rule.
Write 8 1
3 as a cube root.
Evaluate the fourth root of 81.
Connections
The result in Example 1a) can be generalized as follows.
b 1
n = n
__
b for any n . If n is even, b must be greater than or equal to 0.
3.3 Rational Exponents • MHR 173
c)
bm n
n b
m
b m
n b n
Example 3
Apply Exponent Rules
a) x x
xb) y y c) x x
Solution
a) x x
x
x
x
x
x
x
b) y y y y
y y
y
y
y
c) x x x x
x
Rewrite the power with a positive exponent.
Apply the product rule.
Apply the quotient rule.
Apply the power of a power rule.
Write the exponents with common denominators.
Apply the product rule.
Express with a positive exponent.
Connections
Note that 1
y
1 6 can also
be expressed as 1
6
__ y .
Apply the power of a power rule.
Apply the product rule.
174 MHR • Functions 11 • Chapter 3
Example 4
Solve a Problem Involving a Rational Exponent
V S
V Sπ
S
Solution
S V
Vπ
π
V
Key Concepts
nn
b n bn n b n b
mn n m
b bm n
n b
m
m n n b
Communicate Your Understanding
C1 a)
b)
C2 a)
i)
ii)
b)
C3
Step Explanation
x x x x
x
x
x
x
x
A Practise
For help with questions 1 and 2, refer to Example 1.
1.
a) b)
c) d)
2.
a) b)
c) d)
For help with questions 3 and 4, refer to Example 2.
3.
a) b)
c) d)
4.
a) b) c)
d) e) f)
For help with questions 5 and 6, refer to Example 3.
5.
a) x x b) m m c) w
w
d) ab
a be) y f) u v
6.
a) k k b) p
p
c) y d) w
e) x x f) y
3.3 Rational Exponents • MHR 175
176 MHR • Functions 11 • Chapter 3
For help with question 7, refer to Example 4.
7. SV
S V π V
B Connect and Apply 8.
a) A
b) A
c)
i) ii) iii)
d)S
e)
i) ii) iii)
9.
a) V
b) V
c)
i) ii) iii)
10.
a)
A S V B S V
C V S D V S
b)
c)
11. a)
i) ii)
b)
c)
i) ii)
d)
12. Chapter Problem
Tr
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
Connections
The square-cube law is important in many areas of science. It can be used to explain why elephants have a maximum size and why giant insects cannot possibly exist, except in science fiction! Why do you think this is so? Go to the Functions 11 page on the McGraw-Hill Ryerson Web site and follow the links to Chapter 3 to find out more about the square-cube law and its impact on humans and other creatures.
T kr k
T
a) k
b)
c)
13.
a)
T kr r T
b)
Achievement Check
14. h m r m
hr
m
a)
i)
ii)
iii)
b)
c) B m
B
C Extend 15. a)
b)
16.
a)
b)
c)
d)
17. PV
P V
a) V
b)
c)
d)
18. Math Contesta b b a
A B
C D
19. Math Contest
3.3 Rational Exponents • MHR 177
Connections
Johannes Kepler lived in the late 16th and early 17th centuries in Europe. He made some of science’s most important discoveries about planetary motion.
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
178 MHR • Functions 11 • Chapter 3
Properties of Exponential Functions
f x abx ab b a b
Investigate
How can you discover the characteristics of the graph of an exponential function?
A: The Effect of b on the Graph of y = abx
f x x a
1. a)
b)
2.
a)
b) y
c) x
d) interval
i)
ii)
3. b
a) y x
b) b
c) b
Tools
• computer with The Geometer’s Sketchpad®
or
• grid paper
or
• graphing calculator
Connections
It is important to be careful around discarded electrical equipment, such as television sets. Even if the device is not connected to a power source, stored electrical energy may be present in the capacitors.
3.4
interval an unbroken part of the
real number line
is either all of or has one of the following forms: x < a, x > a, x ≤ a, x ≥ a, a < x < b, a ≤ x ≤ b, a < x ≤ b, a ≤ x < b, where a, b , and a < b
3.4 Properties of Exponential Functions • MHR 179
The Geometer’s Sketchpad b
Graph New Parameter bOK
Graph Plot New Functionb x OK
b
bb
b Edit Parameter
b Animate ParameterMotion Controller b
4. b
a)
b) b
c) b
5. b
6. Reflect bf x bx
B: The Effect of a on the Graph of y = abx
f x a x ba
1. a f x x
a) a
b) a
c) a
2. Reflect af x a x
x ya
Connections
In Section 3.3, you saw how to deal with rational exponents. The definition of an exponent can be extended to include real numbers, and so the domain of a function like f (x) = 2x is all real numbers. Try to evaluate 2π using a calculator.
Technology TipWith a graphing calculator, you can vary the line style to see multiple graphs simultaneously. Refer to the Technology Appendix on pages 496 to 516.
180 MHR • Functions 11 • Chapter 3
f x x
xyx x
Example 1
Analyse the Graph of an Exponential Function
x y
a) yx
b) y x
Solution
a) yx
Method 1: Use a Table of Values
xy
x y
—2 16
—1 8
0 4
1 2
2 1
3 1
2
4 1
4
Connections
You encountered horizontal and vertical asymptotes in Chapter 1 Functions.
4 1
2 —2
= 4 2
1 2
= 16
x
12
16
20
—4
y
8
4
2—2 0 4
y = 4( )x12
3.4 Properties of Exponential Functions • MHR 181
Method 2: Use a Graphing Calculator
xx
y yy y
xx
y y
yx
x yx x y
b) y x
x
y y
x
y
x
x y
Connections
Could you use transformations to quickly sketch the graph of y = —3—x ? You will explore this option in Section 3.5.
182 MHR • Functions 11 • Chapter 3
Example 2
Write an Exponential Equation Given Its Graph
y abx
Solution
x y
x y
0 2
1 6
2 18
3 54
by abx
b a
x y b
y abx
a
a
a
y x
x
18
24
30
36
42
48
54y
12
6
2—2 0 4
x
18
24
30
36
42
48
54y
12
6
2—2 0 4
(3, 54)
(2, 18)
(1, 6)
(0, 2)
Change in y
3
3
3
3.4 Properties of Exponential Functions • MHR 183
Example 3
Write an Exponential Function Given Its Properties
a)
b)
Solution
a)
A x Ax
x A
Ax
Start with 200 mg. After every half-life, the amount is reduced by half.
A A xx
A x
A t x t
The half-life of this material is 3 days. Therefore, the number of elapsed half-lives at any given point is the number of days divided by 3.
A tt
At
b)
t
t
A tt
t t
t
120
160
280
80
40
20 4
200
240
A(t) = 200( )12
t3
A
120
160
280
A
80
40
t20 4
200
240
184 MHR • Functions 11 • Chapter 3
Communicate Your Understanding
C1 a)
b) y abx x
C2 f xx
g x x
a)
i) y y ii) y y
b)
i) ii)
C3
Key Concepts
y abx
y abx a b
x y y
y y a
y abx a b
x y y
y y a
y
0 x
x
y
0 x
y
0 x
y
0
ab
ab
ab
ab
3.4 Properties of Exponential Functions • MHR 185
A Practise
For help with questions 1 to 3, refer to Example 1.
1.
a)
x
4
—8
—4
420—4 —2
8
y
b)
2
—4
—2
x42—4
4
y
0—2
c)
x
4
—8
—4
42—4
8
y
0—2
d)
x
4
—8
—4
42—4
8
y
0—2
A y x
B yx
C y x
D y x
2. a)
x
y y
y
b)
3. a)
x
y y
y
b)
For help with questions 4 and 5, refer to Example 2.
4.
x
12
42—4
16
4 (0, 4)
(1, 8)
(2, 16)
8
y
0—2
186 MHR • Functions 11 • Chapter 3
5.
For help with question 6, refer to Example 3.
6.
a)t
A
A At
B At
C At
D At
b)
B Connect and Apply 7.
i)
ii)
iii) x y
iv)
v)
a) f xx
b) y x
c) yx
8. a)
i) f x x
ii) r x x
b)
c)
9. a)
i) g xx
ii) r x x
b)
c)
10. a)
i) f x x ii) g xx
b)
c)
11.
t
a)
b)
c)
d)
e)
24
—8
y
16
8
2—2 0 4
32
48
56
40
x
(0, 24)
(1, 12)(2, 6)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
2
3
V
1
t30 21Time (milliseconds)
Vo
lta
ge
Dro
p (
V)
3.4 Properties of Exponential Functions • MHR 187
12.
R
tR t t
a)
b)
c)
d)
i) ii)
C Extend 13. Use Technology
V V b t
RC V
V tR
C
R C
a) b
b)
c)
d)
e)
14.
a)
b)
c)
15.
a)
b)
c)
d)
16. Math Contestx x
17. Math Contest
yx
y b
x a a b
A B C D
18. Math Contest
A B
C D
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
188 MHR • Functions 11 • Chapter 3
Transformations of Exponential Functions
f x f x c f x f x d
f x af x f x f kx
Investigate
How can you transform an exponential function?
1. a)y x y x
b)
c)
i) y x y x
ii)
d)
2. a) y x y x
b)
c)
3. a) y x
y x
b)
Tools
• computer with The Geometer’s Sketchpad
or
• graphing calculator
3.5
3.5 Transformations of Exponential Functions • MHR 189
4. a) y x y x
b)
c)
5.
6. Reflect
7. a) Reflect y x
b)
Example 1
Horizontal and Vertical Translations
y x
a) y x
b) y x
c) y x
Solution
a) y x
y x
y x
x
y y y
2
—2
x42—4
4
y
0—2
—4
y = 3x
y = 3x — 4
190 MHR • Functions 11 • Chapter 3
b) y x
y x
y x
y
x
y y
c) y x
y x
y
x
y y
Example 2
Stretches, Compressions, and Reflections
y x
a) y x
b) y x
c) y x
2
—2
x42—4
4
y
0—2
—4
y = 3xy = 3x — 2
6
2
4
x42—4
8
y
0—2
y = 3x
y = 3x + 1 + 3
(0, 1)
(—1, 4)
3.5 Transformations of Exponential Functions • MHR 191
Solution
a) y x
y x
b) y x
y x
y x
c) y x
y x
y x
x
12
4
8
x42—4
16
y
0—2
y = 4x
y = 2(4x)
8
—16
—8
x
y = 2(4x)
y = —2(4x)
42—4
16
y
0—2
y x
x
y
12
4
8
x
y = 4x
42—4
16
y
0—2
y = 412 x
12
4
8
x42—4
16
y
0—2
y = 412 xy = 4
12 x—
y
192 MHR • Functions 11 • Chapter 3
Example 3
Graph y = abk(x — d) + c
y x
Solution
y x
y x
y x
y x
Example 4
Circuit Analysis
I tI t t
a)
b) i) ii)
c)
x
Connections
You explored compound transformations of functions in Chapter 2.
12
4
8
x42
16
y
0—2
y = 22x
y = 2x
8
—16
—8
x
y = 22x
y = —22x
42
16
y
0—2
xy x
8
—16
—8
x
y = —22x
y = —22(x — 3) + 5
42
16
y
0—2
3.5 Transformations of Exponential Functions • MHR 193
Solution
Method 1: Use a Graphing Calculator
a)
t t
b) i)
2nd 1:value
ENTER
t
ii)
t
c) t t
Method 2: Use The Geometer’s Sketchpad®
a) GraphPlot New Function
OK
GraphGrid FormRectangular Grid
t t
194 MHR • Functions 11 • Chapter 3
b) i) It
Construct Point On Function Plot
Measure Coordinates
tI
t
ii)
t
c)
t t
Key Concepts
y abk x d cy bx
Horizontal and Vertical Translations
d dd
c cc
Vertical Stretches, Compressions, and Reflections
a a
a a
a xx
yy = abx, a = 1
a = —1
0 < a < 1
—1 < a < 00
a > 1
a < —1
0 x
y
y = bx + c, c = 0
c < 0
c > 0
x
y
d > 0
d < 0
y = bx — d, d = 0
0
3.5 Transformations of Exponential Functions • MHR 195
Key Concepts
Horizontal Stretches, Compressions, and Reflections
kk
kk
k y
y x y x
x
yy = bkx, k = 1k = —1
0 < k < 1—1 < k < 0
k > 1k < —1
0
Communicate Your Understanding
C1 y x
Transformation
a)
b)
c)
d)
e)
f)
g) x
C2
C3 a) y x
b) y x
C4 a k c d y bx
y abk x d c
C5
a)
b)
A y = x
B y = x +
C y = – x
D y = x
E y = x
F y = x
G y = x
196 MHR • Functions 11 • Chapter 3
A Practise
For help with questions 1 to 3, refer to Example 1.
1. y x
a) y x b) y x
c) y x d) y x
2. y x
3.
y x
a)
b)
c)
d)
For help with questions 4 to 6, refer to Example 2.
4. y x
a) y x b) y x
c) y x d) y x
5. y x
6.
y x
a) x
b)
c)
d) y
For help with question 7 and 8, refer to Example 3.
7. y x
y x
8. y x
y x
B Connect and Apply
For help with question 9, refer to Example 4.
9. Tt
T t t
a)
b)
c)
10. a) f x x
b)
i)
ii)
iii)
11. a) y x
b) y x
y x
c) y x
y x
d)
12. a) y x
b)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
13. a)
y y
b)
14. a)
b)
c)
15.
a)
y bx
b)
c)
d)
e)
f)
16.
Term 1 Term 2 Term 3
a)
b)
c)s
n
d)
i)
ii)
17.
a)n
t
b)
c)
d) tn
e) tn
f)
18. Use Technologya
a
a)
b)a
c)
3.5 Transformations of Exponential Functions • MHR 197
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
198 MHR • Functions 11 • Chapter 3
19. Use Technology
d
a)
b)d
c)
Achievement Check
20. a) y x
yx
b) yx
c)
i)
ii)
iii)
iv)
C Extend 21.
a)
b)
c)
d)
22. Use Technology
yx x
a)
b)
c)
d)
e)
f)
23. Math Contest f x x
f x f x f x kf xk
24. Math Contest y
y
A B C D
25. Math Contest
yx
26. Math Contest
A B C D
A B C D
3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 199
3.6
Making Connections: Tools and Strategies for Applying Exponential Models
Example 1
Construct a Model of Exponential Growth
a)
b)
c)
Solution
Method 1: Use a Graphing Calculator
a)
STAT 1:Edit
L1 L2
Y=
Plot1
2nd Plot1
Year Earnings ($)
2002 679
2003 688
2004 702
2005 725
2006 747
200 MHR • Functions 11 • Chapter 3
ZOOM 9:ZoomStat
2nd
STAT CALC0:ExpReg
2nd , 2nd ,
VARS
Y-VARS 1:Function
1:Y1 ENTER
a by abx
y x
x yn E
nE n n
GRAPH
b) E
n
2nd
1:value
c)
y
3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 201
2nd
5:intersect
Method 2: Use Fathom™
a)
y abx
a b
a
b
b
202 MHR • Functions 11 • Chapter 3
Graph Plot Function
OK
n
E
n
E n n
b) n
E n n
E
c)
Graph Moveable Line
Technology TipYou can find parameters a and b under the Global Values heading and Year under the Attributes heading.
Double-click on these headings to show or hide the list of values.
Double-click on the value to enter it into the equation.
3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 203
Example 2
Choose a Model of Depreciation
n
a)
b)
c)
Solution
a) L1 L2 Table Editor
L3 2nd OPS 7: List( 2nd
) ENTER
Number of Years, n Value ($)
1 1200
2 960
3 768
4 614
5 492
6 393
204 MHR • Functions 11 • Chapter 3
b) Linear Model
Y1
v n n v
n
Plot1 Y1
ZOOM 9:ZoomStat
Quadratic Model
Y2
v n n n v
n
Y1Plot1 Y2 ZOOM
9:ZoomStat
ZOOM 3:Zoom Out ENTER
Technology Tip To perform a regression, press
STAT , cursor over to CALC, select the desired type of regression, and enter the appropriate data lists.
To store the function, press VARS , select Y-VARS, and select the desired function location.
Remember to use commas to separate data lists and function variables.
Connections
You may see values of r and/or r2 after performing a regression. These values are related to how well a regression line or curve fits the data. The closer r is to 1 or —1, and the closer r2 is to 1, the better the fit. To enable the viewing of these parameters, press 2nd STAT 0 for [CATALOG] and then scroll down and select Diagnostics On. You will learn more about these parameters if you take Mathematics of Data Management in grade 12.
3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 205
Exponential Model
Y3
v n n v
n
Y1 Y2
Plot1 Y3 ZOOM
9:ZoomStat
ZOOM
3:Zoom Out ENTER
c) n
Method 1: Use the Graph
2nd
1:value
ENTER
Method 2: Use the Equation
v n n
v
206 MHR • Functions 11 • Chapter 3
Key Concepts
Fathom
Communicate Your Understanding
C1
a)
x
y
0
b)
x
y
0
c)
x
y
0
d)
x
y
0
C2 a)
b)
C3 a)
i)
ii)
b)
3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 207
A Practise
For help with questions 1 to 4, refer to Example 1.
1.
a)
12
16
20
—4
y
8
4
x2—2 0 4
b)
12
16
20
—4
y
8
4
x2—2 0 4
c)
12
16
20
—4
y
8
4
x2—2 0 4
2.
3.
95100105110115120
130135
125
V
t30
(0, 100)
(1, 110)
(2, 121)
(3, 133.1)
21Time (years)
Va
lue
($
)
a)
b) a b
V n a bn
c)
d)
e)
A y x
B y x
C y x
D y x
E y x
F y x
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
Connections
Data from investment earnings can look linear, but often involve an exponential relationship. You will learn more about investment and loan calculations in Chapter 7 Financial Applications.
208 MHR • Functions 11 • Chapter 3
For help with questions 4 and 5, refer to Example 2.
4.
a)
b)
c)
d)
B Connect and Apply 5.
a)
b)
6.
a)
b)
c)
d)
7.
8.
a)
b)
c)
d)
9.
Year CPI ($)
2002 100
2003 102.8
2004 104.7
2005 107
2006 109.1
2007 111.8
2008 115.6
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
Year Population
0 800
1 830
2 870
3 900
4 940
5 970
3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 209
a)
b)
10.
a)
b)
c)
d)
11.
a)
b)
12.
a)
t
b)
c)
d)
e)
f)
13.
a)
b)
C Extend 14.
15. Math Contest
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
A
B C
X
Y
Z
3.1 The Nature of Exponential Growth, pages 150 to 157
1.
A P n B P n
C P n D P n
2. a)
b)
c)
d)
e)
3. a)
b)
3.2 Exponential Decay: Connecting to Negative Exponents, pages 160 to 169
4.
a)
b)
c)
5.
a)
b)
6.
a) b) c)
d) e) f)
7.
a) x x x b) km k m
c) w w d) u vu v
e) z f) ab
3.3 Rational Exponents, pages 170 to 177
8.
a) b) c)
d) e) f)
g) h) i)
9. xk
U
x Uk
a)
b)
c)
3.4 Properties of Exponential Functions, pages 178 to 187
10. a) yx
b)
i) ii)
iii) x y
iv)
v)
Chapter 3 Review
210 MHR • Functions 11 • Chapter 3
11.
3.5 Transformations of Exponential Functions, pages 188 to 198
12. a) y x
b)
i)
ii)
iii)
13.
y x
a) y x b) y x
c) y x d) y x
3.6 Making Connections: Tools and Strategies for Applying Exponential Models, pages 199 to 209
14. hn
Number of Bounces, n Height, h (cm)
0 100
1 76
2 57
3 43
4 32
5 24
a)
b)
c)
d)
i)
ii)
e)
120
160
y
80
40
x2—4 —2 0 4(0, 10)
(1, 40)
(2, 160)
Chapter Problem WRAP-UP
a)
b)
Chapter 3 Review • MHR 211
212 MHR • Functions 11 • Chapter 3
For questions 1 to 4, select the best answer.
1. One day, 5 friends started a rumour. They agreed that they would each tell the rumour to two different friends the next day. On each day that followed, every person who just heard the rumour would tell another two people who had not heard the rumour. Which equation describes the relation between the number of days that have elapsed, d, and the number of people, P, who hear the rumour on that day?
A P 5 2 5d B P 5 5 2d
C P 5 5 ( 1 _ 2 ) d D P 5 2 ( 1 _
5 ) d
2. What is the value of 4 1 _ 2
?
A 2 B 1 _ 2 C
1 _ 2 D1 _ 16
3. Which is the correct equation when y 5 5x is translated 3 units down and 4 units left?
A y 5 5x 4 3 B y 5 5x 4 3
C y 5 5x 3 4 D y 5 5x 3 4
4. Which is correct about exponential functions?
A The ratio of successive first differences is constant.
B The first differences are constant.
C The first differences are zero.
D The second differences are constant.
5. Evaluate. Express your answers as integers or fractions in lowest terms.
a) 49 1 _ 2
b) 53 c) (4)0
d) 16 1 _ 4
e) (8) 5 _ 3
f) ( 3 _ 4 ) 4
g) ( 27 _ 64
) 1 _
3
h) ( 8 _ 125
) 4 _
3
6. Simplify. Express your answers using only positive exponents.
a) (x2)(x3)(x4) b)p3
_ p2
c) (2k4)1 d) ( a 1 _ 2
)( a 2 _ 3
)
e) ( y 2 _ 3
)6 f) ( u 1 _ 2
v3)2
7. Build or sketch a model that shows an exponential growing pattern that is doubling from one term to the next.
a) Sketch the first three terms of the model.
b) Make a table of values that relates the number of objects to the term number.
c) Graph the relation.
d) Find the first and second differences.
e) Write an equation for this model.
f) Give at least three reasons that confirm that the pattern is exponential in nature.
8. Match each graph with its corresponding equation.
a)
6
x42—4
8
2
4
y
0—2
b)
6
x42—4
8
2
4
y
0—2
c)
6
x42—4
8
2
4
y
0—2
Chapter 3 Practice Test
d)
6
x42—4
8
2
4
y
—2 0
A y 5 x2
B y 5 2x
C y 5 2x
D y 5 ( 1 _ 2 )
x
9. a) Sketch the graph of the function y 5 2x – 5 3.
b) Identify the
i) domain
ii) range
iii) equation of the asymptote
10. Describe the transformation(s) that map the base function y 5 8x onto each function.
a) y 5 1 _ 3
(8x) b) y 5 84x
c) y 5 –8–x d) y 5 8–3x – 6
11. A radioactive substance with an initial mass of 80 mg has a half-life of 2.5 days.
a) Write an equation to relate the mass remaining to time.
b) Graph the function. Describe the shape of the curve.
c) Limit the domain so that the model accurately describes the situation.
d) Find the amount remaining after
i) 10 days
ii) 15 days
e) How long will it take for the sample to decay to 5% of its initial mass?
12. a) Sketch the function y 5 ( 1 _ 2 ) 2x 3 1
by applying transformations to the graph of the base function y 5 2x.
b) For the transformed function, find the
i) domain
ii) range
iii) equation of the asymptote
13. The height, h, of a square-based pyramid is related to its volume, V, and base side length, b, by the equation h 5 3Vb2. A square-based pyramid has a volume of 6250 m3 and a base side length of 25 m. Find its height.
14. The population of Pebble Valley over a period of 5 years is shown.
Year Population
0 2000
1 2150
2 2300
3 2500
4 2700
5 2950
a) Make a scatter plot for this relationship. Do the data appear to be exponential in nature? Explain your reasoning.
b) Find the equation of the curve of best fit.
c) Limit the domain of the function so that it accurately models this situation.
d) Predict the population of Pebble Valley 7 years after the first table entry. State any assumptions you must make.
e) How long will it take for the town to double in size? State any assumptions you must make.
Chapter 3 Practice Test • MHR 213
218 MHR • Functions 11 • Chapter 3
Radioactive Isotopes
Radioactive isotopes are used in nuclear medicine to allow physicians to explore bodily structures in patients. Very small quantities of these isotopes, such as iodine-131 (I-131; half-life 8.065 days), are injected into patients. The isotopes are traced through the body so a medical diagnosis can be made. These elements have a very short half-life, so the label on the bottle will state the radioactivity of the element (the amount of the element that remains) at a particular moment in time.
a) A bottle of I-131 is delivered to a hospital on September 5. It states on the label that the radioactivity at 12:00 p.m. on September 10 will be 380 megabecquerels (MBq).
i) Write a defining equation for the relationship between the amount of radioactivity and the time since the bottle was delivered.
ii) What is the radioactive activity at
• 12:00 p.m. on the delivery day?
• 6:00 a.m. on September 25?
iii)Sketch the graph of this relationship.
b) Effective half-life is the time required for a radioactive isotope contained in a body to reduce its radioactivity by half, due to a combination of radioactive decay and the natural elimination of the isotope from the body. In patients with Graves disease, the effective half-life of I-131 is 62.5% of the normal half-life. In patients with a disease called toxic nodular goitre, the effective half-life is 75% of the normal half-life. Compare these results to the standard rate of decay for I-131 numerically, graphically, and algebraically.
c) Research one of the topics below and write a short report on your findings.
• other radioactive isotopes that are used in medical diagnoses or treatments
• diseases that can be treated with radioactive isotopes• the production and transport of radioactive isotopes• the training required for medical personnel to use radioactive isotopes
Task