# Analytic Geometry

## Conic Sections in Polar Coordinates

Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets’ orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.
In an elliptical orbit, the periapsis is the point at which the two objects are closest, and the apoapsis is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body’s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.

### Identifying a Conic in Polar Form

Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph. Consider the parabola
       shown in .
In The Parabola, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus
       at the pole, and a line, the directrix, which is perpendicular to the polar axis.
If
       is a fixed point, the focus, and
       is a fixed line, the directrix, then we can let
       be a fixed positive number, called the eccentricity, which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points
       such that
       is a conic. In other words, we can define a conic as the set of all points
       with the property that the ratio of the distance from
       to
       to the distance from
       to
       is equal to the constant

For a conic with eccentricity

1. if
     the conic is an ellipse
2. if
     the conic is a parabola
3. if
     the conic is an hyperbola
With this definition, we may now define a conic in terms of the directrix,
       the eccentricity
       and the angle
       Thus, each conic may be written as a polar equation, an equation written in terms of
       and

###  Graphing the Polar Equations of Conics

When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine
       and, therefore, the shape of the curve. The next step is to substitute values for
       and solve for
       to plot a few key points. Setting
       equal to
       and
       provides the vertices so we can create a rough sketch of the graph.

###  Deﬁning Conics in Terms of a Focus and a Directrix

So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.

### Key Concepts

1. Any conic may be determined by a single focus, the corresponding eccentricity,  and the directrix. We can also define a conic in terms of a fixed point, the focus
         at the pole, and a line, the directrix, which is perpendicular to the polar axis.
2. A conic is the set of all points
         where eccentricity
         is a positive real number. Each conic may be written in terms of its polar equation. See .
3. The polar equations of conics can be graphed. See , , and .
4. Conics can be defined in terms of a focus, a directrix, and eccentricity. See  and .
5. We can use the identities
  and
  to convert the equation for a conic from polar to rectangular form. See .

### Section Exercises

### Verbal

###  Algebraic

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.
For the following exercises, convert the polar equation of a conic section to a rectangular equation.
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

###  Extensions

Recall from Rotation of Axes that equations of conics with an
         term have rotated graphs. For the following exercises, express each equation in polar form with
         as a function of

### Chapter Review Exercises

### The Ellipse

For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.
For the following exercises, graph the ellipse, noting center, vertices, and foci.
For the following exercises, use the given information to find the equation for the ellipse.

### The Hyperbola

For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.
For the following exercises,  graph the hyperbola, labeling vertices and foci.
For the following exercises, find the equation of the hyperbola.

### The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.
For the following exercises, graph the parabola, labeling vertex, focus, and directrix.
For the following exercises, write the equation of the parabola using the given information.

### Rotation of Axes

For the following exercises, determine which of the conic sections is represented.
For the following exercises, determine the angle
         that will eliminate the
         term, and write the corresponding equation without the
         term.
For the following exercises, graph the equation relative to the
         system in which the equation has no
         term.

### Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.
For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

### Practice Test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.
For the following exercises, sketch the graph, identifying the center, vertices, and foci.
For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.
For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.
For the following exercises, determine which conic section is represented by the given equation, and then determine the angle
       that will eliminate the
       term.
For the following exercises, rewrite in the
       system without the
       term, and graph the rotated graph.
For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.
For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.


# Sequences, Probability, and Counting Theory

## Introduction to Sequences, Probability and Counting Theory

A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a new home? A luxury convertible? A cruise around the world?
The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over an extended period of time.
This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum.
In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning.


# Sequences, Probability, and Counting Theory

## Sequences and Their Notations

### Learning Objectives

1. Write the first few terms of a sequence (IA 12.1.1)
2. Find a formula for the general term (nth term) of a sequence (IA 12.1.2)

### Objective 1: Write the first few terms of a sequence (IA 12.1.1).

A patient takes a 30 mg antibiotic capsule. At the end of that hour, the amount of antibiotic remaining in her body is only 90% of the amount in the beginning of that hour. The 30mg dose is taken at time t = 1 hour. How much of this dose remains at the end of 1 hour? 2hours? 3 hours? 4 hours?
This ordered list of numbers 27, 24.3, 21.87, 19.68, … is a sequence. Each number in the list is a term.

A sequence is a function whose domain is the counting numbers. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an infinite sequence.
Often when working with sequences we do not want to write out all the terms. We want a more compact way to show how each term is defined. When we worked with functions, we wrote  and we said the expression 2x was the rule that defined values in the range.

While a sequence is a function, we do not use the usual function notation. Instead of writing the function as  , we would write it as  . The  is the , the term in the nth position where n is a value in the domain. The formula for writing the nth term of the sequence is called the general term or formula of the sequence.

General sequence terms are denoted as follows:

### Practice Makes Perfect

Write the first few terms of a sequence.

### Objective 2: Find a formula for the general term (nth term) of a sequence (IA 12.1.2)

Sometimes we have a few terms of a sequence and it would be helpful to know the general term or . To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.

### Practice Makes Perfect

A video game company launches an exciting new advertising campaign. They predict the number of online visits to their website, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the third day, and so on. See .
If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of ordered lists.

### Writing the Terms of a Sequence Defined by an Explicit Formula

One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is
The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term. The first five terms of this sequence are 2, 4, 8, 16, and 32.
Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.
One type of formula is an explicit formula, which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We can use the formula to find the nth term of the sequence, where
  is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the
  term.

The first term of the sequence is
  the second term is
  the third term is
  and so on. The
  term of the sequence can be found by raising 2 to the
  power. An explicit formula for a sequence is named by a lower case letter
  with the subscript
  The explicit formula for this sequence is
Now that we have a formula for the
  term of the sequence, we can answer the question posed at the beginning of this section. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To find the number of hits on the last day of the month, we need to find the 31st term of the sequence. We will substitute 31 for
  in the formula.
If the doubling trend continues, the company will get
  hits on the last day of the month. That is over 2.1 billion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions.
Another way to represent the sequence is by using a table. The first five terms of the sequence and the
  term of the sequence are shown in .
Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in  that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function.
Lastly, we can write this particular sequence as
A sequence that continues indefinitely is called an infinite sequence. The domain of an infinite sequence is the set of counting numbers. If we consider only the first 10 terms of the sequence, we could write
This sequence is called a finite sequence because it does not continue indefinitely.

### Investigating Alternating Sequences

Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as
  increases. Let’s take a look at the following sequence.
Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence.

### Investigating Piecewise Explicit Formulas

We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions. Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.

### Finding an Explicit Formula

Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the
  term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.

### Writing the Terms of a Sequence Defined by a Recursive Formula

Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.
Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms.
A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining
  in terms of preceding terms. For example, suppose we know the following:
We can find the subsequent terms of the sequence using the first term.
So the first four terms of the sequence are
  .
The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.
To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so

### Using Factorial Notation

The formulas for some sequences include products of consecutive positive integers.  , written as
  is the product of the positive integers from 1 to
  For example,
An example of formula containing a factorial is
  The sixth term of the sequence can be found by substituting 6 for

The factorial of any whole number
  is
  We can therefore also think of
  as

### Key Equations

### Key Concepts

1. A sequence is a list of numbers, called terms, written in a specific order.
2. Explicit formulas define each term of a sequence using the position of the term. See , , and .
3. An explicit formula for the
  term of a sequence can be written by analyzing the pattern of several terms. See .
4. Recursive formulas define each term of a sequence using previous terms.
5. Recursive formulas must state the initial term, or terms, of a sequence.
6. A set of terms can be written by using a recursive formula. See  and .
7. A factorial is a mathematical operation that can be defined recursively.
8. The factorial of
  is the product of all integers from 1 to
  See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, write the first four terms of the sequence.
For the following exercises, write the first eight terms of the piecewise sequence.
For the following exercises, write an explicit formula for each sequence.
For the following exercises, write the first five terms of the sequence.
For the following exercises, write the first eight terms of the sequence.
For the following exercises, write a recursive formula for each sequence.
For the following exercises, evaluate the factorial.
For the following exercises, write the first four terms of the sequence.

### Graphical

For the following exercises, graph the first five terms of the indicated sequence
For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.
For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

### Technology

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

For the following exercises, use the steps above to find the indicated term or terms for the sequence.
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

### Extensions


# Sequences, Probability, and Counting Theory

## Arithmetic Sequences

### Learning Objectives

1. Determine if a sequence is arithmetic (IA 12.2.1)
2. Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)

### Objective 1: Determine if a sequence is arithmetic (IA 12.2.1)

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

The difference between consecutive terms, d, and is called the common difference, for n greater than or equal to two.

### Practice Makes Perfect

Determine if each sequence is arithmetic. If so, indicate the common difference.

### Practice Makes Perfect

### Objective 2: Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)

In the last section, we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is  and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with  .
The first term adds 0d to the  , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to  is one less than the number of the term. We then have the formula for the general term of an arithmetic sequence.

### Practice Makes Perfect

Find the general term (nth term) of an arithmetic sequence.

Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
As an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.

### Finding Common Differences

The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.
The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.

### Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of
 and
 into formula below.

### Using Recursive Formulas for Arithmetic Sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

### Using Explicit Formulas for Arithmetic Sequences

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

The common difference is
 , so the sequence represents a linear function with a slope of
 . To find the
 -intercept, we subtract
 from
  . You can also find the
 -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in .
Recall the slope-intercept form of a line is
  When dealing with sequences, we use
  in place of
  and
  in place of
  If we know the slope and vertical intercept of the function, we can substitute them for
  and
  in the slope-intercept form of a line. Substituting
  for the slope and
  for the vertical intercept, we get the following equation:
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is
 , which simplifies to

### Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

### Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term of
 instead of
 In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

### Key Equations

### Key Concepts

1. An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
2. The constant between two consecutive terms is called the common difference.
3. The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See .
4. The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See  and .
5. A recursive formula for an arithmetic sequence with common difference
  is given by
  See .
6. As with any recursive formula, the initial term of the sequence must be given.
7. An explicit formula for an arithmetic sequence with common difference
  is given by
  See .
8. An explicit formula can be used to find the number of terms in a sequence. See .
9. In application problems, we sometimes alter the explicit formula slightly to
  See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the common difference for the arithmetic sequence provided.
For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.
For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.
For the following exercises, write the first five terms of the arithmetic series given two terms.
For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.
For the following exercises, find the first term given two terms from an arithmetic sequence.
For the following exercises, find the specified term given two terms from an arithmetic sequence.
For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.
For the following exercises, write a recursive formula for each arithmetic sequence.
For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.
For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.
For the following exercises, write an explicit formula for each arithmetic sequence.
For the following exercises, find the number of terms in the given finite arithmetic sequence.

### Graphical

For the following exercises, determine whether the graph shown represents an arithmetic sequence.
For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.

### Technology

For the following exercises, follow the steps to work with the arithmetic sequence
  using a graphing calculator:

For the following exercises, follow the steps given above to work with the arithmetic sequence
  using a graphing calculator.

### Extensions


# Sequences, Probability, and Counting Theory

## Geometric Sequences

### Learning Objectives

1. Determine if a sequence is geometric (IA 12.3.1).
2. Find the general term (nth term) of a geometric sequence (IA 12.3.2).

### Objective 1: Determine if a sequence is geometric (IA 12.3.1)

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same.

The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.

### Practice Makes Perfect

Determine if each sequence is geometric. If so, indicate the common ratio.

### Practice Makes Perfect

### Objective 2: Find the general term (nth term) of a geometric sequence (IA 12.3.2)

Let’s find the formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is  and the common ratio is  . We will then look for a pattern.

### Practice Makes Perfect

Find the general term (nth term) of a geometric sequence.

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.

### Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

### Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is
  and the common ratio is
  we can find subsequent terms by multiplying
  to get
  then multiplying the result
  to get
  and so on.
The first four terms are

### Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

### Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
Let’s take a look at the sequence
  This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is
The graph of the sequence is shown in .

### Solving Application Problems with Geometric Sequences

In real-world scenarios involving geometric sequences, we may need to use an initial term of
  instead of
  In these problems, we can alter the explicit formula slightly by using the following formula:

### Key Equations

### Key Concepts

1. A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
2. The constant ratio between two consecutive terms is called the common ratio.
3. The common ratio can be found by dividing any term in the sequence by the previous term. See .
4. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See  and .
5. A recursive formula for a geometric sequence with common ratio
  is given by
  for
  .
6. As with any recursive formula, the initial term of the sequence must be given. See .
7. An explicit formula for a geometric sequence with common ratio
  is given by
  See .
8. In application problems, we sometimes alter the explicit formula slightly to
  See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the common ratio for the geometric sequence.
For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.
For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.
For the following exercises, write the first five terms of the geometric sequence, given any two terms.
For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.
For the following exercises, find the specified term for the geometric sequence, given the first four terms.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write a recursive formula for each geometric sequence.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write an explicit formula for each geometric sequence.
For the following exercises, find the specified term for the geometric sequence given.
For the following exercises, find the number of terms in the given finite geometric sequence.

### Graphical

For the following exercises, determine whether the graph shown represents a geometric sequence.
For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

### Extensions


# Sequences, Probability, and Counting Theory

## Series and Their Notations

### Learning Objectives

1. Use summation notation to write a sum. (IA 12.1.5)
2. Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Objective 1: Use summation notation to write a sum. (IA 12.1.5)

A series is the sum of the terms of a sequence. For example,  1 + 6 + 11+ 16 + 21 + 26 + 31 is the sum of the first seven terms arithmetic sequence with general term,
We write a series by using the summation notation. In order to write that summation, we will need to find the general term of our sequence and the summation will look like:

For the series, 1 + 6 + 11 + 16  + 21 + 26 + 31 + .... the summation notation is

### Practice Makes Perfect

Use summation notation to write the sum.

### Objective 2: Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Practice Makes Perfect

A parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.

### Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.
The  of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation
 represents the partial sum.
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma,
  to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series
  for
  through
  We can begin by substituting the terms for
 and listing out the terms of this series.
We can find the sum of the series by adding the terms:

### Using the Formula for Arithmetic Series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,
 The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first
 terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first
  terms of an arithmetic series, we can derive a formula for the sum of the first
  terms of any arithmetic series.
Because there are
 terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first
  terms of an arithmetic series.
Use the formula to find the sum of each arithmetic series.

### Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio,
  We can write the sum of the first
  terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first
  terms of a geometric series. We will begin by multiplying both sides of the equation by

Next, we subtract this equation from the original equation.
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for
 divide both sides by

Use the formula to find the indicated partial sum of each geometric series.

### Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first
 terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is

This series can also be written in summation notation as
 where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

### Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined.  The terms in this series approach 0:
The common ratio

 As
  gets very large, the values of
 get very small and approach 0.   Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with
 approach 0; the sum of a geometric series is defined when

Determine whether the sum of the infinite series is defined.

### Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first
 terms of a geometric series.
We will examine an infinite series with
 What happens to
  as
  increases?
The value of
  decreases rapidly. What happens for greater values of

As
  gets very large,
 gets very small. We say that, as
  increases without bound,
 approaches 0. As
  approaches 0,
 approaches 1. When this happens, the numerator approaches
  This give us a formula for the sum of an infinite geometric series.
Find the sum, if it exists.

### Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with
 and
 After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after
 deposits using the formula for the sum of the first
 terms of a geometric series. In 6 years, there are 72 months, so
 We can substitute
 into the formula, and simplify to find the value of the annuity after 6 years.
After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of $50 each for a total of
  This means that because of the annuity, the parent earned $720.44 interest in their college fund.

### Key Equations

### Key Concepts

1. The sum of the terms in a sequence is called a series.
2. A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See .
3. The sum of the terms in an arithmetic sequence is called an arithmetic series.
4. The sum of the first
 terms of an arithmetic series can be found using a formula. See  and .
5. The sum of the terms in a geometric sequence is called a geometric series.
6. The sum of the first
 terms of a geometric series can be found using a formula. See  and .
7. The sum of an infinite series exists if the series is geometric with

8. If the sum of an infinite series exists, it can be found using a formula. See , , and .
9. An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, express each description of a sum using summation notation.
For the following exercises, express each arithmetic sum using summation notation.
For the following exercises, use the formula for the sum of the first
 terms of each arithmetic sequence.
For the following exercises, express each geometric sum using summation notation.
For the following exercises, use the formula for the sum of the first
  terms of each geometric sequence, and then state the indicated sum.
For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

### Graphical

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.
For the following exercises, use the geometric series

### Numeric

For the following exercises, find the indicated sum.
For the following exercises, use the formula for the sum of the first  terms of an arithmetic series to find the sum.
For the following exercises, use the formula for the sum of the first
  terms of a geometric series to find the partial sum.
For the following exercises, find the sum of the infinite geometric series.
For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

### Extensions

### Real-World Applications


# Sequences, Probability, and Counting Theory

## Counting Principles

### Learning Objectives

1. Solve counting problems using the addition principle.
2. Solve counting problems using the multiplication principle.

### Objective 1: Solve counting problems using the addition principle.

In probability theory, an outcome is a possible result of an experiment or trial.

In probability theory, an event is a set of outcomes of an experiment.

Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.

The addition principle applies when we are making only one selection.

### Practice Makes Perfect

Solve counting problems using the addition principle.

### Objective 2: Solve counting problems using the multiplication principle.

The Multiplication Principle applies when we are making more than one selection.

### Practice Makes Perfect

Solve counting problems using the multiplication principle.

A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.

### Using the Addition Principle

The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in .

### Using the Multiplication Principle

The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are 2 appetizer options, 3 entrée options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in .
The possible choices are:
1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding
7. salad, chicken, cake
8. salad, chicken, pudding
9. salad, fish, cake
10. salad, fish, pudding
11. salad, steak, cake
12. salad, steak, pudding
We can also find the total number of possible dinners by multiplying.
We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.

### Finding the Number of Permutations of n Distinct Objects

The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.

### Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle

To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.

There are four options for the first place, so we write a 4 on the first line.

After the first place has been filled, there are three options for the second place so we write a 3 on the second line.

After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.

There are 24 possible permutations of the paintings.
A family of five is having portraits taken. Use the Multiplication Principle to find the following.

### Finding the Number of Permutations of n Distinct Objects Using a Formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of
  objects and we want to choose
  objects from the set in order, we write
  Another way to write this is
  a notation commonly seen on computers and calculators. To calculate
  we begin by finding
  the number of ways to line up all
  objects. We then divide by
  to cancel out the
  items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is
  Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all
  objects and placing them in order. In that case we would be dividing by
  or
  which we said earlier is equal to 1. So the number of permutations of
  objects taken
  at a time is
  or just

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

### Find the Number of Combinations Using the Formula

So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of
  objects from a set of
  objects where the order does not matter can be written as
  Just as with permutations,
  can also be written as
  In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are
  ways to order 3 paintings. There are
  or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.

### Finding the Number of Subsets of a Set

We have looked only at combination problems in which we chose exactly
  objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is
  way to order a pizza with no toppings. There are
  ways to order a pizza with exactly one topping. If we continue this process, we get
There are 32 possible pizzas. This result is equal to

We are presented with a sequence of choices. For each of the
  objects we have two choices: include it in the subset or not. So for the whole subset we have made
  choices, each with two options. So there are a total of
  possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.

### Finding the Number of Permutations of n Non-Distinct Objects

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be  ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the
  permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are
  ways to order the stars and
  ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.

### Key Equations

### Key Concepts

1. If one event can occur in
  ways and a second event with no common outcomes can occur in
  ways, then the first or second event can occur in
  ways. See .
2. If one event can occur in
  ways and a second event can occur in
  ways after the first event has occurred, then the two events can occur in
  ways. See .
3. A permutation is an ordering of
  objects.
4. If we have a set of
  objects and we want to choose
  objects from the set in order, we write

5. Permutation problems can be solved using the Multiplication Principle or the formula for
  See  and .
6. A selection of objects where the order does not matter is a combination.
7. Given
  distinct objects, the number of ways to select
  objects from the set is
  and can be found using a formula. See .
8. A set containing
  distinct objects has
  subsets. See .
9. For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See .

### Section Exercises

### Verbal

For the following exercises, assume that there are
  ways an event
  can happen,
  ways an event
  can happen, and that
  are non-overlapping.
Answer the following questions.

### Numeric

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
For the following exercises, compute the value of the expression.
For the following exercises, find the number of subsets in each given set.
For the following exercises, find the distinct number of arrangements.

### Extensions

### Real-World Applications


# Sequences, Probability, and Counting Theory

## Binomial Theorem

### Learning Objectives

1. Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

### Objective 1: Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

Pascal’s triangle helps us find the coefficients of the terms in the expansion of a binomial.

To find the coefficients of the terms, we write our expansion again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients. The array to the right is called Pascal’s Triangle.

Notice that in each expansion the powers of a in each term decrease from n to 0, and the powers of b increase from 0 to n.

Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.

To find the coefficients of the expansion of the binomial  , go to the row that has the value n as a second entry.

### Practice Makes Perfect

Use Pascal’s Triangle to expand a binomial.

A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find
  without multiplying the binomial by itself
  times.

### Identifying Binomial Coefficients

In Counting Principles, we studied combinations. In the shortcut to finding
  we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation
  instead of
  but it can be calculated in the same way. So
The combination
  is called a binomial coefficient. An example of a binomial coefficient is

### Using the Binomial Theorem

When we expand
  by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand
  we might multiply
  by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for
  decreases and the exponent for
  increases. The sum of the two exponents is
  for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form,
  to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial

See , which illustrates the following:
1. There are
  terms in the expansion of

2. The degree (or sum of the exponents) for each term is

3. The powers on
  begin with
  and decrease to 0.
4. The powers on
  begin with 0 and increase to

5. The coefficients are symmetric.
To determine the expansion on
  we see
  thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of
  the pattern is as follows:
1. Introduce
  and then for each successive term reduce the exponent on
  by 1 until
  is reached.
2. Introduce
  and then increase the exponent on
  by 1 until
  is reached.

The next expansion would be
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in . Pascal didn't invent the triangle. The underlying principles had been developed and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first published in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is named in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a number of probability problems.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add
  to find the middle number, 2. In the
  row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.

### Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such as
  can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of

The second term is
  The third term is
  We can generalize this result.

### Key Equations

### Key Concepts

1.
  is called a binomial coefficient and is equal to
  See .
2. The Binomial Theorem allows us to expand binomials without multiplying. See .
3. We can find a given term of a binomial expansion without fully expanding the binomial. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the binomial coefficient.
For the following exercises, use the Binomial Theorem to expand each binomial.
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

### Graphical

For the following exercises, use the Binomial Theorem to expand the binomial
  Then find and graph each indicated sum on one set of axes.

### Extensions


# Sequences, Probability, and Counting Theory

## Probability

### Learning Objectives

1. Introduction to Sample Spaces and Computing Basic Probabilities.

### Objective 1: Introduction to Sample Spaces and Computing Basic Probabilities.

Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.

The probability of an event is a description of how likely it is that an event will happen. A probability is a number between 0 and 1 (that is, between 0% and 100%), where probabilities closer to 100% are very likely to occur, and probabilities closer to 0% are very unlikely to occur. A probability of 0% means the event is impossible, and a probability of 100% means the event will certainly occur.

A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.

The sample space S for a probability model is the set of all possible outcomes. For example, the sample space for rolling a dice is the set 1,2,3,4,5,6.This notation is referred to as roster notation.

An event A is a subset of the sample space S. For example, the event “Rolling an even number” is the subset 2,4,6.

To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.

It is important to note that in order to use this formula, all outcomes must be equally likely to happen.

For example, the probability of rolling an even number with a standard dice is:

### Practice Makes Perfect

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

### Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is  An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event
 is a number that always satisfies
  where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like .
The sum of the probabilities listed in a probability model must equal 1, or 100%.

### Computing Probabilities of Equally Likely Outcomes

Let
 be a sample space for an experiment. When investigating probability, an event is any subset of
 When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in
 Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in
 so the probability of the event is

### Computing the Probability of the Union of Two Events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events
  is the event that occurs if either or both events occur.
Suppose the spinner in  is spun. We want to find the probability of spinning orange or spinning a

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is
  There are a total of 6 sections, and 2 of them have a
 So the probability of spinning a
 is
  If we added these two probabilities, we would be counting the sector that is both orange and a
 twice. To find the probability of spinning an orange or a
 we need to subtract the probability that the sector is both orange and has a

The probability of spinning orange or a
 is

### Computing the Probability of Mutually Exclusive Events

Suppose the spinner in  is spun again, but this time we are interested in the probability of spinning an orange or a
 There are no sectors that are both orange and contain a
 so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is
Notice that with mutually exclusive events, the intersection of
 and
 is the empty set. The probability of spinning an orange is
 and the probability of spinning a
 is
 We can find the probability of spinning an orange or a
 simply by adding the two probabilities.
The probability of spinning an orange or a
 is

### Using the Complement Rule to Compute Probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event
 denoted
  is the set of outcomes in the sample space that are not in
 For example, suppose we are interested in the probability that a horse will lose a race. If event
 is the horse winning the race, then the complement of event
 is the horse losing the race.
To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is
 the probability of the horse losing the race is simply

### Computing Probability Using Counting Theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are
 ways to select 2 phones that are not defective. There are 8 phones, so there are
 ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

### Key Equations

### Key Concepts

1. Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
2. The probabilities in a probability model must sum to 1. See .
3. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See .
4. To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See .
5. To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See .
6. The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See .
7. In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, use the spinner shown in  to find the probabilities indicated.
For the following exercises, two coins are tossed.
For the following exercises, four coins are tossed.
For the following exercises, one card is drawn from a standard deck of
  cards. Find the probability of drawing the following:
For the following exercises, two dice are rolled, and the results are summed.
For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
For the following exercises, use this scenario: a bag of M&Ms contains
  blue,
  brown,
  orange,
  yellow,
  red, and
  green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

### Extensions

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting
  numbers from the numbers
  to
  After the player makes his selections,
  winning numbers are randomly selected from numbers
  to
  A win occurs if the player has correctly selected
  or
  of the
  winning numbers. (Round all answers to the nearest hundredth of a percent.)

### Real-World Applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov

### Chapter Review Exercises

### Sequences and Their Notation

### Arithmetic Sequences

### Geometric Sequences

### Series and Their Notation

### Counting Principles

### Binomial Theorem

### Probability

For the following exercises, assume two die are rolled.
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

### Practice Test

For the following exercises, use the spinner in .


# Functions

## Introduction to Functions

Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.
Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.


# Functions

## Functions and Function Notation

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

### Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
The domain is

 The range is

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter
    Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter

A function
  is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain,
  is paired with exactly one element in the range,

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain,

 is not paired with exactly one element in the range,

For example, the term “odd” corresponds to three values from the range,

 and the term “even” corresponds to two values from the range,

 This violates the definition of a function, so this relation is not a function.
 compares relations that are functions and not functions.

### Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
To represent “height is a function of age,” we start by identifying the descriptive variables

 for height and

 for age. The letters
  and
  are often used to represent functions just as we use
  and
  to represent numbers and

 and
  to represent sets.

Remember, we can use any letter to name the function; the notation
   shows us that
   depends on
  The value
   must be put into the function
   to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
We can also give an algebraic expression as the input to a function. For example
   means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

### Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function
   where
   identifies months by an integer rather than by name.
 defines a function
   Remember, this notation tells us that
   is the name of the function that takes the input
   and gives the output

 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

### Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

### Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function
     can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

### Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation
     expresses a functional relationship between

    and
  We can rewrite it to decide if
      is a function of

### Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See .http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.
At times, evaluating a function in table form may be more useful than using equations.  Here let us call the function

 The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function
     at the input value of “goldfish.” We would write
  Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function
     seems ideally suited to this function, more so than writing it in paragraph or function form.

### Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

### Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.
However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in .
This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.
To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b). The function in part (a) shows a relationship that is not a one-to-one function because inputs
    and
    both give output
    The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

### Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.
The most common graphs name the input value
    and the output value
    and we say
    is a function of
    or
    when the function is named
    The graph of the function is the set of all points
    in the plane that satisfies the equation
    If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in  tell us that
    and
    However, the set of all points
    satisfying
    is a curve. The curve shown includes
    and
    because the curve passes through those points.
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See .

###  Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

### Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use
    as the input variable and
    as the output variable.
We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in .

### Key Equations

### Key Concepts

1. A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See  and .
2. Function notation is a shorthand method for relating the input to the output in the form
     See  and .
3. In tabular form, a function can be represented by rows or columns that relate to input and output values. See .
4. To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See  and .
5. To solve for a specific function value, we determine the input values that yield the specific output value. See .
6. An algebraic form of a function can be written from an equation. See  and .
7. Input and output values of a function can be identified from a table. See .
8. Relating input values to output values on a graph is another way to evaluate a function. See .
9. A function is one-to-one if each output value corresponds to only one input value. See .
10. A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See .
11. The graph of a one-to-one function passes the horizontal line test. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine whether the relation represents
   as a function of

For the following exercises, evaluate
   at the indicated values

### Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
For the following exercises, determine if the given graph is a one-to-one function.

### Numeric

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine if the relation represented in table form represents
   as a function of

For the following exercises, use the function
   represented in .
For the following exercises, evaluate the function
   at the values
   and

For the following exercises, evaluate the expressions, given functions
   and

1.

2.

3.

### Technology

For the following exercises, graph
  on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
   on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
   on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
   on the given domain. Determine the corresponding range. Show each graph.

###  Real-World Applications


# Functions

## Domain and Range

Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring.  shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

### Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.
We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See .
We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write
 We will discuss interval notation in greater detail later.
Let’s turn our attention to finding the domain of a function whose equation is provided.  Oftentimes, finding the domain of such functions involves remembering three different forms.  First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers.  Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero.  Third, if there is an even root, consider excluding values that would make the radicand negative.
Before we begin, let us review the conventions of interval notation:
1. The smallest term from the interval is written first.
2. The largest term in the interval is written second, following a comma.
3. Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
4. Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.
See  for a summary of interval notation.

### Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example,
 describes the behavior of
 in set-builder notation. The braces
 are read as “the set of,” and the vertical bar | is read as “such that,” so we would read
 as “the set of x-values such that 10 is less than or equal to
 and
 is less than 30.”
 compares inequality notation,  set-builder notation, and interval notation.
To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol,
 to combine two unconnected intervals. For example, the union of the sets

and

is the set
 It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

### Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See .
We can observe that the graph extends horizontally from
 to the right without bound, so the domain is
 The vertical extent of the graph is all range values
  and below, so the range is
 Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

### Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

### Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function
 With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.
If we input 0, or a positive value, the output is the same as the input.
If we input a negative value, the output is the opposite of the input.
Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.
We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income
 would be
 if
  and
  if

### Key Concepts

1. The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
2. The domain of a function can be determined by listing the input values of a set of ordered pairs. See .
3. The domain of a function can also be determined by identifying the input values of a function written as an equation. See , , and .
4. Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See .
5. For many functions, the domain and range can be determined from a graph. See  and .
6. An understanding of toolkit functions can be used to find the domain and range of related functions. See , , and .
7. A piecewise function is described by more than one formula. See  and .
8. A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the domain of each function using interval notation.

### Graphical

For the following exercises, write the domain and range of each function using interval notation.
For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

### Numeric

For the following exercises, given each function
 evaluate
 and

For the following exercises, given each function
 evaluate
 and

For the following exercises, write the domain for the piecewise function in interval notation.

### Technology

###  Extension

###  Real-World Applications