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Unpacking the Standards
Understanding the standards and the vocabulary terms in the standards will
help you know exactly what you are expected to learn in this chapter.
CC.9-12.F.BF.2
Write arithmetic and geometric
sequences both recursively and with
an explicit formula, use them to model
situations, and translate between the
two forms.
arithmetic sequence (sucesión
aritmética) A sequence whose
successive terms differ by the same
nonzero number d, called the
common difference.
geometric sequence (sucesión
geométrica) A sequence in which
the ratio of successive terms is a
constant r, called the common ratio,
where r ≠ 0 and r ≠ 1.
recursive formula (fórmula
recurrente) A formula for a
sequence in which one or more
previous terms are used to generate
the next term.
CC.9-12.F.IF.7e
Graph exponential … functions,
showing intercepts and end
behavior, …
Lesson 9-1
You can write rules for arithmetic and geometric sequences as a
function of the term number or with respect to the previous term. You
can use the form that is more useful for a particular situation.
EXAMPLE
Explicit and Recursive Formulas
In the geometric sequence below, each term is twice the previous term.
So, the common ratio is r = 2.
1
2
3
4
Position, n
3
6
12
24
Term, ​an​ ​
​a1​ ​
​a​2​
​a​3​
​a​4​
Explicit formula: ​an​ ​= ​a1​ ​​r​n - 1​, so ​an​ ​= 3 • ​2n​ - 1​
Recursive formula: The recursive formula gives the first term and for
finding successive terms:
​a​n​= ​an - 1
​ ​r, so ​a1​ ​= 3, ​an​ ​= ​2a​n - 1​
What It Means For You
Lessons 9-2, 9-3
The graph of an exponential function f​(x)​= ​ab​x​has y-intercept a.
If a > 0, the function may model growth or decay.
EXAMPLE
Key Vocabulary
50
exponential function (función
exponencial) A function of the form
f​(x)​= ​ab​x​, where a and b are real
numbers with a ≠ 0, b > 0, and
b ≠ 1.
y
y
8
40
6
y = 3(0.5)x
30
20
y = 3(4)x
(0, 3)
2
10
x
(0, 3)
-4
-2
0
2
x
492
-2
2
4
4
The graph nears the x-axis as x
decreases and rises faster and
faster as x increases.
Chapter 9
-4
0
The graph nears the x-axis as
x increases and rises faster and
faster as x decreases.
Exponential Functions
© Houghton Mifflin Harcourt Publishing Company
chapter 9
Key Vocabulary
What It Means For You
What It Means For You
CC.9-12.F.LE.1
Distinguish between situations that can
be modeled with linear functions and
with exponential functions.
Key Vocabulary
linear function (función lineal) A
function that can be written in the
form y = mx + b, where x is the
independent variable and m and b
are real numbers. Its graph is a line.
Lesson 9-5
A linear function models a constant amount of change for equal
intervals. An exponential function models a constant factor, or
constant ratio of change for equal intervals.
EXAMPLE
Exponential model
Value of a car
Car's Age (yr)
0
1
2
3
+1
+1
+1
NON-EXAMPLE
Ratio is
constant.
Value ($)
20,000
17,000
14,450
12,282.50
× 0.85
× 0.85
× 0.85
Nonlinear, non-exponential model
Cable's Distance
from Tower (ft)
Cable's
Height (ft)
0
100
200
300
400
256
144
64
+ 100
+ 100
+ 100
- 144, × 0.64
- 112, × 0.56
- 80, × 0.44
Neither difference nor ratio is constant.
CC.9-12.F.LE.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).
What It Means For You
Lessons 9-1, 9-2, 9-3
You can construct a model of a linear or exponential function from
different descriptions or displays of the same situation.
EXAMPLE
Geometric Sequence
A ball is dropped 81 inches onto a hard surface. The table shows the
ball’s height on successive bounces. Write a model for the height
reached as a function of the number of bounces.
Bounce
1
2
3
4
Height (in.)
54
36
24
16
Consecutive terms have a common ratio of __
​ 23 ​.  You can write a model as
an exponential function or as a geometric sequence:
x
()
Exponential function: f​(x)​= 81​​ __
​ 23 ​  ​​ ​, where x is the bounce
number
Geometric sequence: a
​ ​1​= 54, ​an​ ​= __
​ 23 ​ ​an - 1
​ ​, where n is the bounce
number
Chapter 9
493
Exponential Functions
chapter 9
© Houghton Mifflin Harcourt Publishing Company; Photo credit: © Jupiterimages/Getty Images
Height of Bridge Suspension Cables
Key Vocabulary
common ratio (razón común) In a geometric sequence, the constant ratio of any term and the
previous term.
exponential decay (decremento exponencial) An exponential function of the form f(x)= ​ab​x​in
which 0 < b < 1. If r is the rate of decay, then the function can be written y = a(1 - r)t , where a is the
initial amount and t is the time.
exponential function (función exponencial) A function of the form f(x)= ​ab​x​, where a and b are
real numbers with a ≠ 0, b > 0, and b ≠ 1.
exponential growth (crecimiento exponencial) An exponential function of the form f(x)= ​ab​x​in
which b > 1. If r is the rate of growth, then the function can be written y = a(1 + r)t, where a is the initial
amount and t is the time.
geometric sequence (sucesión geométrica) A sequence in which the ratio of successive terms is a
constant r, called the common ratio, where r ≠ 0 and r ≠ 1.
linear function (función lineal) A function that can be written in the form y = mx + b, where x is the
independent variable and m and b are real numbers. Its graph is a line.
chapter 9
recursive formula (fórmula recurrente) A formula for a sequence in which one or more previous
terms are used to generate the next term.
The Common Core Standards for Mathematical Practice describe varieties
of expertise that mathematics educators at all levels should seek to develop
in their students. Opportunities to develop these practices are integrated
throughout this program.
1. M
ake sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. C
onstruct viable arguments and critique the reasoning of others.
4. M
odel with mathematics.
5. Use appropriate tools strategically.
6. A
ttend to precision.
7. L ook for and make use of structure.
8. L ook for and express regularity in repeated reasoning
Chapter 9
494
Exponential Functions