Alg1 power points_-_unit_7_-_exponents_and_exponential_functions
1. Unit Essential Questions
How can you simplify expressions involving exponents?
What are the key features and essential components of exponential
functions?
2. MACC.912.A-SSE.A.2: Use the structure of an expression to
identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to
interpret expressions for exponential functions.
4. KEY CONCEPTS AND
VOCABULARY
PRODUCT OF POWERS
PROPERTY
POWER OFA POWER
PROPERTY
For every nonzero number a and integers
m and n,
Example:
For every nonzero number a and
integers m and n,
Example:
POWER OF A PRODUCT PROPERTY
For every nonzero numbers a and b and integer m,
Example:
am
×an
= am+n
am
( )
n
= am×n
x2
×x8
= x2+8
= x10 x3
( )
6
= x3×6
= x18
ab( )m
= am
bm
2x( )4
= 24
x4
=16x4
7. EXAMPLE 3:
SIMPLIFYING POWER OF A PRODUCT
Simplify.
a)
b)
c)
-3xy( )2
a2
b9
( )
4
2x2
yz3
( )
5
9x2
y2
a8
b36
32x10
y5
z15
8. Express the area as a monomial.
a) b)
EXAMPLE 4:
SIMPLIFYING USING GEOMETRIC FORMULAS
r = xyz2
h = x3
y2
b = 2x2
y
px2
y2
z4
x5
y3
9. Simplify.
a)
b)
c)
EXAMPLE 5:
SIMPLIFYING MORE CHALLENGING
EXPONENTIAL EXPRESSIONS
5
6
x3æ
èç
ö
ø÷
2
4y3
( ) 3
4
xy4æ
èç
ö
ø÷ -3x2
y2
( )
3x2
( )
2
2xy( )2
é
ë
ù
û
3
25
36
x6
-9x3
y9
576x10
y6
10. RATE YOUR UNDERSTANDING
MULTIPLICATION PROPERTIES OF
EXPONENTS
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to interpret expressions for exponential
functions.
RATING LEARNING SCALE
4
I am able to
• simplify expressions using the multiplication properties of
exponents in more challenging problems that I have never
previously attempted
3
I am able to
• simplify expressions using the multiplication properties of
exponents
2
I am able to
• simplify expressions using the multiplication properties of
exponents with help
1
I am able to
• identify the multiplications properties of exponents
TARGET
11. MACC.912.A-SSE.A.2: Use the structure of an expression to
identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to
interpret expressions for exponential functions.
13. KEY CONCEPTS AND
VOCABULARY
QUOTIENT OF POWERS
PROPERTY
POWER OFA QUOTIENT
PROPERTY
For every nonzero number a and integers m and n,
Example:
For every nonzero numbers a and b and integer m,
Example:
ZERO EXPONENT PROPERTY NEGATIVE EXPONENT PROPERTY
For every nonzero number a,
Example:
For every nonzero number a and integer n,
Example:
am
an = am-n
a
b
æ
èç
ö
ø÷
m
=
am
bm
a0
=1 a-n
=
1
an
x5
x2 = x5-2
= x3 x
2
æ
èç
ö
ø÷
4
=
x4
24
=
x4
16
2xyz
p
æ
èç
ö
ø÷
0
= 1 3-4
=
1
34
18. Simplify.
a)
b)
c)
EXAMPLE 5:
SIMPLIFYING MORE CHALLENGING
EXPONENTIAL EXPRESSIONS
16x2
y-1
( )
0
4x0
y-4
z( )
-3
80
c2
d3
f
4c-3
d-4
æ
èç
ö
ø÷
-2
3x3
y2
( )
3
6x2
y-3
( )
-2
64z3
y12
16
c10
d14
f 2
972x13
19. RATE YOUR UNDERSTANDING
DIVISION PROPERTIES OF EXPONENTS
MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.F-IF.C.8b: Use the properties of exponents to interpret expressions for exponential
functions.
RATING LEARNING SCALE
4
I am able to
• divide expressions using the properties of exponents for
more challenging problems that I have never previously
attempted
3
I am able to
• divide expressions using the properties of exponents
• simplify expressions containing negative and zero exponents
2
I am able to
• divide expressions using the properties of exponents with
help
• simplify expressions containing negative and zero exponents
with help
1
I am able to
• understand the division properties of exponents
TARGET
20. MACC.912.N-RN.A.1: Explain how the definition of the
meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for
a notation for radicals in terms of rational exponents.
MACC.912.N-RN.A.2: Rewrite expressions involving
radicals and rational exponents using the properties of
exponents.
22. KEY CONCEPTS AND
VOCABULARY
RATIONAL EXPONENTS
If the nth root of a b is a real number and m and n are
positive integers, then
anda
1
n = an a
m
n = an
( )
m
24. Convert to exponential form.
a)
b)
c)
d)
EXAMPLE 2:
CONVERTING TO EXPONENTIAL FORM
x4
a5
b35
(2x)73
x
1
4
a
1
5
b
3
5
2
7
3 x
7
3
25. EXAMPLE 3:
CONVERTING TO RADICAL FORM
Convert to radical form.
a) b)
c) d)
t
1
2 x
3
7
3x
3
2 (4a)
3
5
t x37
3 x3
(4a)35
26. Evaluate.
a) b)
c) d)
EXAMPLE 4:
EVALUATING AN EXPRESSION WITH A
RATIONAL EXPONENT
64
1
6 8
2
3
1
81
æ
èç
ö
ø÷
1
4
625
3
4
2 4
1
3
125
27. Solve.
a) b)
c) d)
EXAMPLE 5:
SOLVING EXPONENTIAL EQUATIONS BY
REWRITING IN EXPONENTIAL FORM
8x
= 64 3x
= 27
3x
= 243 12x
=144
2 3
5 2
28. The frequency f in hertz of the nth key on a piano is .
If a middle C is the 40th key, what is the frequency of a middle C?
EXAMPLE 6:
MODELING EXPONENTIAL EXPRESSIONS IN
REAL-WORLD SITUATIONS
f = 440 2
1
2
æ
è
ç
ö
ø
÷
n-49
19.45 hertz
29. RATE YOUR UNDERSTANDING
RATIONAL EXPONENTS
MACC.912.N-RN.A.1: Explain how the definition of the meaning of rational exponents follows from extending
the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational
exponents.
MACC.912.N-RN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
RATING LEARNING SCALE
4
I am able to
• evaluate and rewrite expressions involving radicals and rational
exponents in real-world situations or more challenging problems
that I have never previously attempted
3
I am able to
• evaluate and rewrite expressions involving radicals and rational
exponents
• solve equations involving expressions with rational exponents
2
I am able to
• evaluate and rewrite expressions involving radicals and rational
exponents with help
• solve equations involving expressions with rational exponents with
help
1
I am able to
• understand that I can use rational exponents to represent radicals
TARGET
30. MACC.912.F-IF.C.7e: Graph exponential and logarithmic functions,
showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
MACC.912.F-LE.A.2: Construct linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs. (include
reading these from a table).
MACC.912.F-IF.C.9: Compare properties of two functions each
represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions).
32. KEY CONCEPTS AND
VOCABULARY
EXPONENTIAL FUNCTIONS
If a ≠ 0 and b >0, then exponential functions are of the
form
y = abx
Notice: the variable x is an exponent
33. Does the table represent a linear or an exponential function?
Explain your reasoning.
a) b)
EXAMPLE 1:
IDENTIFYING LINEAR AND EXPONENTIAL
FUNCTIONS FROM A TABLE OF VALUES
x 1 2 3 4
y 3 6 12 24
x 1 2 3 4
y 10 13 16 19
Exponential
Common ratio of 2
Linear
Common difference of 3
34. Is the function linear or exponential? Explain your
reasoning.
a) b)
EXAMPLE 2:
IDENTIFYING LINEAR AND EXPONENTIAL
FUNCTIONS GIVEN A FUNCTION RULE
y = 3x + 4 y = 4
1
3
æ
èç
ö
ø÷
x
Linear
Equation is in
slope-intercept form
Exponential
Variable is in the exponent
35. EXAMPLE 3:
EVALUATING AN EXPONENTIAL FUNCTION
Evaluate for the given value.
a)
b)
c)
f (x) =15×(2)x
; f (3)
f (x) = -10×
1
3
æ
èç
ö
ø÷
x
; f (-1)
f (x) = 200×(5)x
; f (-2)
120
–30
8
36. Graph the exponential function.
a) b)
EXAMPLE 4:
GRAPHING AN EXPONENTIAL FUNCTION
f (x) = 2×(3)x
f (x) = 4 ×
1
2
æ
èç
ö
ø÷
x
37. Determine if the graph is exponential, linear, or neither.
a) b) c)
EXAMPLE 5:
IDENTIFYING AN EXPONENTIAL GRAPH
Linear Neither Exponential
38. Order the functions from least to greatest for f(100).
a) b) c)
EXAMPLE 6:
COMPARING LINEAR GROWTH TO
EXPONENTIAL GROWTH
f(x)
f (x) = 3x
x 1 2 3 4
y 2 4 8 16
3 – Lowest value
at f(100)
1 – Highest value
at f(100)
2 – Middle value
at f(100)
39. RATE YOUR UNDERSTANDING
EXPONENTIAL FUNCTIONS
MACC.912.F-IF.C.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.
MACC.912.F-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences,
given a graph, a description of a relationship, or two input-output pairs. (include reading these from a table).
MACC.912.F-IF.C.9: Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
RATING LEARNING SCALE
4
I am able to
• evaluate and graph exponential functions in real-world situations or
more challenging problems that I have never previously attempted
3
I am able to
• evaluate and graph exponential functions
• identify data and graphs that represents exponential behavior
2
I am able to
• evaluate and graph exponential functions with help
• identify data and graphs that represents exponential behavior with
help
1
I am able to
• understand that exponential functions are non-linear and model an
initial amount that is multiplied by the same number
TARGET
40. MACC.912.F-IF.C.8b: Use the properties of exponents to interpret
expressions for exponential functions.
MACC.912.F-LE.A.1a: Prove that linear functions grow by equal
differences over equal intervals, and that exponential functions grow by
equal factors over equal intervals.
MACC.912.F-LE.A.3: Observe using graphs and tables that a quantity
increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
MACC.912.A-SSE.B.3c: Use the properties of exponents to transform
expressions for exponential functions.
41. WARM UP
Explain why each expression is not in simplest form.
1)
2)
3)
4)
53
x2
x4
y-2
x-1
y2
z0
(2x)3
53 = 125
Negative Exponent
Negative Exponent
z0 = 1
Distribute Power
42. KEY CONCEPTS AND
VOCABULARY
EXPONENTIAL GROWTH AND DECAY
y = abx when b is between 0 and 1
Example: Graph y = 100(0.5)x
Exponential Decay
y = abx when b>1
Example: Graph y = 1(2)x
Exponential Growth
Exponential decay model:
a = initial amount, r = rate, t = time
Exponential growth model:
a = initial amount, r = rate, t = time
y = a(1- r)t
y = a(1+ r)t
Compound Interest – the interest earned or paid on both
the initial investment and previously earned interest.
43. Without graphing, determine whether the function represents
exponential growth or decay. Then find the y-intercept.
a) b) c) y = 0.455(3)x
EXAMPLE 1:
IDENTIFYING EXPONENTIAL GROWTH OR
DECAY
y = 3×
2
3
æ
èç
ö
ø÷
x
y =
1
3
×(2)x
Decay Growth Growth
y-intercept =
(0, 3)
y-intercept =
(0, 1/3)
y-intercept =
(0, 0.455)
44. Write an exponential function to model each situation. Find
each amount after the specified time.
a) A population of 2,155,000 grows 1.3% per year for 10
years.
b) A population of 824,000 decreases 1.4% per year for 18
years.
c) A new car that sells for $27,000 depreciates 25% each
year for 4 years.
EXAMPLE 2:
WRITING AND SOLVING EXPONENTIAL
MODELS
y = 2,155,000(1+ 0.013)10
y = 2,452,120
y = 824,000(1- 0.014)18
y = 639,310
y = 27,000(1- 0.25)4
y = $8542.97
45. EXAMPLE 3:
REAL-WORLD APPLICATIONS OF EXPONENTIAL
GROWTH AND DECAY
a) You invested $1000 in a savings account at the end of 6th
grade. The account pays 5% annual interest. How much
money will be in the account after 6 years?
b) Each year the local country club sponsors a tennis
tournament. Play starts with 128 participants. During
each round, half of the players are eliminated. How
many players remain after 5 rounds?
$1340.10
4 Players
46. A mother and father were negotiating an allowance with their
child. They offered him two scenarios that he could choose from.
The first scenario offered the child $20 every week. The second
scenario offered $0.01 the first week and the amount would
double every week after. Which scenario should the son choose?
Explain by graphing each function.
EXAMPLE 4:
COMPARING EXPONENTIAL GROWTH AND
LINEAR GROWTH
The son should choose the $0.01 option.
After around 15 weeks, the allowance is
significantly more than the linear $20 a
week option.
47. RATE YOUR UNDERSTANDING
EXPONENTIAL GROWTH AND DECAY
MACC.912.F-IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions.
MACC.912.F-LE.A.1a: Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
MACC.912.F-LE.A.3: Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
MACC.912.A-SSE.B.3c: Use the properties of exponents to transform expressions for exponential functions.
RATING LEARNING SCALE
4
I am able to
• model exponential growth and decay in real-world situations
or more challenging problems that I have never previously
attempted
3
I am able to
• model exponential growth and decay
2
I am able to
• model exponential growth and decay with help
1
I am able to
• identify growth and decay looking at a model
TARGET
48. MACC.912.F-IF.A.3: Recognize that sequences are functions,
sometimes defined recursively, whose domain is a subset of the integers.
For example, the Fibonacci sequence is defined recursively by f(0) =
f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
MACC.912.F-BF.A.2: Write arithmetic and geometric sequences both
recursively and with an explicit formula, use them to model situations,
and translate between the two forms.
MACC.912.F-LE.A.1c: Recognize situations in which a quantity grows
or decays by a constant percent rate per unit interval relative to
another.
49. WARM UP
Describe the pattern. Find the next 2 terms in each
sequence.
1) 2, 4, 8, 16,… 2) -–1, –4, –16, –64
Multiply 2 to each terms
32, 64
Multiply 4 to each term
–256, –1024
50. KEY CONCEPTS AND
VOCABULARY
Geometric Sequence – has a common ratio, r, between a term and its
preceding term that is constant.
Common Ratio - the name of the ratio in a Geometric Sequence.
16, 8, 4, 2, 1,… is a geometric sequence with
First term: a1 = 16
Common ratio: r = 1/2
Explicit Formula
The nth term of a geometric sequence with first term a1 and common ratio
r is given by:
an = a1rn – 1, for n > 1
51. For the following sequences, describe the patterns and identify
the next 3 terms.
a) 1, 4, 16, 48,… b) 2, –4, 8, –16,…
EXAMPLE 1:
EXTENDING GEOMETRIC SEQUENCES
Multiply 4 to each term
192, 768, 3072
Multiply –2 to each term
32, –64, 128
52. For the following sequences, identify whether it is a geometric
sequence. If it is, find the common ratio.
a) 4, –8, 16, –32,… b) 3, 9, –27, –81, 243,…
EXAMPLE 2:
IDENTIFYING A GEOMETRIC SEQUENCE
Yes, r = –2 No
53. EXAMPLE 3:
WRITING AN EQUATION FOR A GEOMETRIC
SEQUENCE
Write an equation for the nth term of the sequence. Then
find a6.
a) 5, 2, 0.8, 0.32,… b) –2, –6, –18,…
an = 5(0.4)n-1
a6 = 0.0512
an = -2(3)n-1
a6 = -486
54. Identify if the sequence is arithmetic, geometric, or neither.
Explain.
a) 1/2, 1/4, 1/8,…
b) 2, –4, 8, 16, –32,…
c) 15, 12, 9, 6,…
EXAMPLE 4:
CLASSIFYING THE SEQUENCE
Geometric
Has a common ration of 1/2
Neither
No Common ratio or difference
Arithmetic
Common difference of –3
55. RATE YOUR UNDERSTANDING
GEOMETRIC SEQUENCES
MACC.912.F-IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) =
f(n) + f(n-1) for n ≥ 1.
MACC.912.F-BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula,
use them to model situations, and translate between the two forms.
MACC.912.F-LE.A.1c: Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
RATING LEARNING SCALE
4
I am able to
• use an explicit formula for a geometric sequence to solve
real world problems or more challenging problems that I
have never previously attempted
3
I am able to
• identify and apply geometric sequences
2
I am able to
• identify and apply geometric sequences with help
1
I am able to
• define a geometric sequence
TARGET
