# Systems of Equations and Inequalities

## Solving Systems with Gaussian Elimination

### Learning Objectives

1. Use row operations on a matrix (IA 4.5.2)
2. Solve systems of equations using matrices (IA 4.5.3)

### Objective 1: Use row operations on a matrix (IA 4.5.2)

In the last section, we learned how to write the augmented matrix for a system of equations.

Once a system of equations is in its augmented matrix form, we will solve by elimination by performing operations on the rows that will lead us to the solution. Our goal will be to get 1 on the diagonal of the matrix and all entries below the diagonal must be zeros.

### Row Operations

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

1. Interchange any two rows.
2. Multiply a row by any real number except 0.
3. Add a nonzero multiple of one row to another row.
These actions are called row operations and will help us use the matrix to solve a system of equations.

### Practice Makes Perfect

### Objective 2: Solve systems of equations using matrices (IA 4.5.3)

To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, the augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and solve for at least one variable. We then substitute this value in another equation to continue to solve for the other variables.

### Practice Makes Perfect

Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables. In this section, we will revisit this technique for solving systems, this time using matrices.

### Writing the Augmented Matrix of a System of Equations

A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following
  system of equations.
We can write this system as an augmented matrix:
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.
A three-by-three system of equations such as
has a coefficient matrix
and is represented by the augmented matrix
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form
  so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.

### Writing a System of Equations from an Augmented Matrix

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.

### Performing Row Operations on a Matrix

Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
1. In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
2. Any all-zero rows are placed at the bottom on the matrix.
3. Any leading 1 is below and to the right of a previous leading 1.
4. Any column containing a leading 1 has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
1. Interchange rows. (Notation:
  )
2. Multiply a row by a constant. (Notation:
  )
3. Add the product of a row multiplied by a constant to another row. (Notation:

Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.

### Solving a System of Linear Equations Using Matrices

We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and back-substitution to obtain row-echelon form. Now, we will take row-echelon form a step farther to solve a 3 by 3 system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.

### Key Concepts

1. An augmented matrix is one that contains the coefficients and constants of a system of equations. See .
2. A matrix augmented with the constant column can be represented as the original system of equations. See .
3. Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
4. We can use Gaussian elimination to solve a system of equations. See , , and .
5. Row operations are performed on matrices to obtain row-echelon form. See .
6. To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See  and .
7. A calculator can be used to solve systems of equations using matrices. See .
8. Many real-world problems can be solved using augmented matrices. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, write the augmented matrix for the linear system.
For the following exercises, write the linear system from the augmented matrix.
For the following exercises, solve the system by Gaussian elimination.

### Extensions

For the following exercises, use Gaussian elimination to solve the system.

### Real-World Applications

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.


# Systems of Equations and Inequalities

## Solving Systems with Inverses

### Learning Objectives

1. Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
2. Evaluate the determinant of a 3x3 matrix (IA 4.6.2)

### Objective 1: Evaluate the determinant of a 2×2 matrix (IA 4.6.1)

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant.

### Practice Makes Perfect

Find the determinant of the 2x2 matrices.

### Objective 2: Evaluate the determinant of a 3×3 matrix (IA 4.6.2)

To evaluate the determinant of a 3×3 matrix, we must be able to evaluate the minor of an entry in the determinant.

The minor of an entry is the 2×2 determinant found by eliminating the row and column in the 3×3 determinant that contains the entry.

For example, to find the minor of entry a1, we eliminate the row and column which contain it. So, we eliminate the first row and first column. Then we write the 2×2 determinant that remains.

To find the minor of entry b2, we eliminate the row and column that contain it. So, we eliminate the second row and second column. Then we write the 2×2 determinant that remains.

### Strategy for evaluating the determinant of a 3x3 matrix

To evaluate a 3×3 determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.

When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.

### Expanding by minors along the first row to evaluate a 3x3 determinant.

To evaluate a 3×3 determinant by expanding by minors along the first row, we use the following pattern:

NOTE: We can evaluate the determinant of a matrix by expanding minors along any row or column. When a row or a column has a zero entry, expanding by that row or column results in less calculations.

### Practice Makes Perfect

Soriya plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Soriya invest in each bond? What is the best method to solve this problem?
There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.

### Finding the Inverse of a Matrix

We know that the multiplicative inverse of a real number
       is
       and
       For example,
       and
       The multiplicative inverse of a matrix is similar in concept, except that the product of matrix
       and its inverse
       equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by
       where
       represents the dimension of the matrix. Observe the following equations.
The identity matrix acts as a 1 in matrix algebra. For example,

A matrix that has a multiplicative inverse has the properties
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as the reversibility,
       is a requirement. Not all square matrices have an inverse, but if
       is invertible, then
       is unique. We will look at two methods for finding the inverse of a
       matrix and a third method that can be used on both
       and
  matrices.

### Finding the Multiplicative Inverse Using Matrix Multiplication

We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.

### Finding the Multiplicative Inverse by Augmenting with the Identity

Another way to find the multiplicative inverse is by augmenting with the identity. When matrix
         is transformed into
         the augmented matrix
         transforms into

For example, given
augment
         with the identity
Perform row operations with the goal of turning
         into the identity.
1. Switch row 1 and row 2.

2. Multiply row 2 by
  and add to row 1.

3. Multiply row 1 by
  and add to row 2.

4. Add row 2 to row 1.

5. Multiply row 2 by

The matrix we have found is

### Finding the Multiplicative Inverse of 2×2 Matrices Using a Formula

When we need to find the multiplicative inverse of a
         matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If
         is a
         matrix, such as
the multiplicative inverse of
         is given by the formula
where
         If
         then
         has no inverse.

### Finding the Multiplicative Inverse of 3×3 Matrices

Unfortunately, we do not have a formula similar to the one for a
         matrix to find the inverse of a
         matrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a

        matrix
augment
         with the identity matrix
To begin, we write the augmented matrix with the identity on the right and
         on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.

### Solving a System of Linear Equations Using the Inverse of a Matrix

Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices:
       is the matrix representing the variables of the system, and
       is the matrix representing the constants. Using matrix multiplication, we may define a system of equations with the same number of equations as variables as
To solve a system of linear equations using an inverse matrix, let
       be the coefficient matrix, let
       be the variable matrix, and let
       be the constant matrix. Thus, we want to solve a system
       For example, look at the following system of equations.
From this system, the coefficient matrix is
The variable matrix is
And the constant matrix is

    Then
       looks like
Recall the discussion earlier in this section regarding multiplying a real number by its inverse,
  To solve a single linear equation
       for
       we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of
       Thus,
The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.
We will investigate this idea in detail, but it is helpful to begin with a
  system and then move on to a
  system.

### Key Equations

### Key Concepts

1. An identity matrix has the property
       See .
2. An invertible matrix has the property
       See .
3. Use matrix multiplication and the identity to find the inverse of a
       matrix. See .
4. The multiplicative inverse can be found using a formula.  See .
5. Another method of finding the inverse is by augmenting with the identity. See .
6. We can augment a
       matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See .
7. 	Write the system of equations as
       and multiply both sides by the inverse of
       See  and .
8. We can also use a calculator to solve a system of equations with matrix inverses. See .

### Section Exercises

### Verbal

### Algebraic

In the following exercises, show that matrix
         is the inverse of matrix

For the following exercises, find the multiplicative inverse of each matrix, if it exists.
For the following exercises, solve the system using the inverse of a
  matrix.
For the following exercises, solve a system using the inverse of a

        matrix.

### Technology

For the following exercises, use a calculator to solve the system of equations with matrix inverses.

### Extensions

For the following exercises, find the inverse of the given matrix.

### Real-World Applications

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.


# Systems of Equations and Inequalities

## Solving Systems with Cramer's Rule

### Learning Objectives

1. Use Cramer’s Rule to solve systems of equations (IA 4.6.3)

### Objective 1: Use Cramer’s Rule to solve systems of equations (IA 4.6.3)

Cramer’s Rule uses determinants to solve systems of equations.

### Practice Makes Perfect

### Practice Makes Perfect

We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.

### Evaluating the Determinant of a 2×2 Matrix

A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.

### Using Cramer’s Rule to Solve a System of Two Equations in Two Variables

We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.
To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.
We eliminate one variable using row operations and solve for the other. Say that we wish to solve for
  If equation (2) is multiplied by the opposite of the coefficient of
  in equation (1), equation (1) is multiplied by the coefficient of
  in equation (2), and we add the two equations, the variable
  will be eliminated.
Now, solve for

Similarly, to solve for
  we will eliminate

Solving for
  gives
Notice that the denominator for both
  and
  is the determinant of the coefficient matrix.
We can use these formulas to solve for
  and
  but Cramer’s Rule also introduces new notation:
1.
  determinant of the coefficient matrix
2.
  determinant of the numerator in the solution of

3.
  determinant of the numerator in the solution of

The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express
  and
  as a quotient of two determinants.

### Evaluating the Determinant of a 3 × 3 Matrix

Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
Find the determinant of the 3×3 matrix.
1. Augment
  with the first two columns.

2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
The algebra is as follows:

### Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. As the order of the matrix increases to 3 × 3, however, there are many more calculations required.
When we calculate the determinant to be zero, Cramer’s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.
Consider a 3 × 3 system of equations.
where
If we are writing the determinant
  we replace the
  column with the constant column. If we are writing the determinant
  we replace the
  column with the constant column. If we are writing the determinant
  we replace the
  column with the constant column. Always check the answer.

### Understanding Properties of Determinants

There are many properties of determinants. Listed here are some properties that may be helpful in calculating the determinant of a matrix.

### Key Concepts

1. The determinant for
  is
  See .
2. Cramer’s Rule replaces a variable column with the constant column. Solutions are
  See .
3. To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See .
4. To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution:
  See .
5. Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See  and .
6. Certain properties of determinants are useful for solving problems. For example:

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the determinant.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
For the following exercises, solve the system of linear equations using Cramer’s Rule.

### Technology

For the following exercises, use the determinant function on a graphing utility.

### Real-World Applications

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.
For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.
For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in .

### Review Exercises

### Systems of Linear Equations: Two Variables

For the following exercises, determine whether the ordered pair is a solution to the system of equations.
For the following exercises, use substitution to solve the system of equations.
For the following exercises, use addition to solve the system of equations.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Systems of Linear Equations: Three Variables

For the following exercises, solve the system of three equations using substitution or addition.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.
For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities.

### Partial Fractions

For the following exercises, decompose into partial fractions.

### Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

### Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.
For the following exercises, write the augmented matrix from the system of linear equations.
For the following exercises, solve the system of linear equations using Gaussian elimination.

### Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.
For the following exercises, find the solutions by computing the inverse of the matrix.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Solving Systems with Cramer's Rule

For the following exercises, find the determinant.
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

### Practice Test

Is the following ordered pair a solution to the system of equations?
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
For the following exercises, graph the following inequalities.
For the following exercises, write the partial fraction decomposition.
For the following exercises, perform the given matrix operations.
For the following exercises, use Gaussian elimination to solve the systems of equations.
For the following exercises, use the inverse of a matrix to solve the systems of equations.
For the following exercises, use Cramer’s Rule to solve the systems of equations.
For the following exercises, solve using a system of linear equations.


# Analytic Geometry

## Introduction to Analytic Geometry

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry.
It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief.
It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.
Other objects in the solar system (and perhaps other systems) follow a similar elliptical path, including the spectacular rings of Saturn. Using this understanding as a basis, 19th century mathematicians like James Clerk Maxwell and Sofya Kovalevskaya showed that despite their appearance through the telescopes of the day (and even in current telescopes), the rings are not solid and continuous, but are rather composed of small particles. Even after the Voyager and Cassini missions have provided close-up and detailed data regarding the ring structures, full understanding of their construction relies heavily on mathematical analysis. Of particular interest are the influences of Saturn's moons and moonlets, and the ways they both disrupt and preserve the ring structure.
In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems.


# Analytic Geometry

## The Ellipse

### Learning Objectives

1. Complete the square of a binomial expression. (IA 9.2.1)
2. Graph a circle. (IA 11.1.4)

### Objective 1: Complete the square of a binomial expression. (IA 9.2.1)

But what happens if we have to solve an equation where the trinomial is not a perfect square?
For example,  ? For these types of equations, we can use a process called completing the square.
Recall  .
We can use the Binomial Squares Pattern to make a perfect square.

### Practice Makes Perfect

Determine what number would have to be added to the given terms to create a perfect square trinomial. Then rewrite as a binomial squared.

### Objective 2: Graph a circle. (IA 11.1.4)

A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, (h, k) and the fixed distance is called the radius, r, of the circle.

The standard or graphing form of the equation of a circle with center, (h, k) and radius, r, is  .
The general form of the equation of a circle is  . If we are given an equation in general form, we can change it to standard, also called the graphing form, by completing the squares in both x  and y. Then we can graph the circle using its center and radius.

### Practice Makes Perfect

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in , is such a room.Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014. It is an semi-circular room called a whispering chamber because the shape makes it possible for sound to travel along the walls and dome. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

### Writing Equations of Ellipses in Standard Form

A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in .
Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points
  in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See .
Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See .
In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.
To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

### Deriving the Equation of an Ellipse Centered at the Origin

To derive the equation of an ellipse centered at the origin, we begin with the foci
  and
  The ellipse is the set of all points
  such that the sum of the distances from
  to the foci is constant, as shown in .
If
  is a vertex of the ellipse, the distance from
  to
  is
  The distance from
  to
  is
  . The sum of the distances from the foci to the vertex is
If
  is a point on the ellipse, then we can define the following variables:
By the definition of an ellipse,
  is constant for any point
  on the ellipse. We know that the sum of these distances is
  for the vertex
  It follows that
  for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.
Thus, the standard equation of an ellipse is
  This equation defines an ellipse centered at the origin. If
  the ellipse is stretched further in the horizontal direction, and if
  the ellipse is stretched further in the vertical direction.

### Writing Equations of Ellipses Centered at the Origin in Standard Form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.
The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

### Writing Equations of Ellipses Not Centered at the Origin

Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated
  units horizontally and
  units vertically, the center of the ellipse will be
  This translation results in the standard form of the equation we saw previously, with
  replaced by
  and y replaced by

###  Graphing Ellipses Centered at the Origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form
  for horizontal ellipses and
  for vertical ellipses.

### Graphing Ellipses Not Centered at the Origin

When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,
    we use the standard forms
    for horizontal ellipses and
    for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

### Solving Applied Problems Involving Ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See . In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.  When these chambers are placed in unexpected places, such as the ones inside Bush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions among travelers.

### Key Equations

### Key Concepts

1. An ellipse is the set of all points
   in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
2. When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See  and .
3. When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See  and .
4. When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See  and .
5. Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.
For the following exercises, find the foci for the given ellipses.

### Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.
For the following exercises, use the given information about the graph of each ellipse to determine its equation.
For the following exercises, given the graph of the ellipse, determine its equation.

###  Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula

###  Real-World Applications


# Analytic Geometry

## The Hyperbola

### Learning Objectives

1. Use the Distance Formula. (IA 11.1.1)
2. Graph a hyperbola with center at (0,0). (IA 11.4.1)

### Objective 1: Use the Distance Formula. (IA 11.1.1)

### Practice Makes Perfect

Use the Distance Formula.

### Objective 2: Graph a hyperbola with center at (0,0). (IA 11.4.1)

A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Notice that, unlike the equation of an ellipse, the denominator of  is not always  and the denominator of  is not always  .
Notice that when the  term is positive, the transverse axis is on the x-axis. When the  term is positive, the transverse axis is on the y-axis.

### Practice Makes Perfect

 Graph a hyperbola with center at (0,0).

What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of conic. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom. See .
Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.

### Locating the Vertices and Foci  of a Hyperbola

In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. See .
Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points
  in a plane such that the difference of the distances between
  and the foci is a positive constant.
Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.
As with the ellipse, every hyperbola has two axes of symmetry. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. See .
In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the x- and y-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.

### Deriving the Equation of a Hyperbola Centered at the Origin

Let
  and
  be the foci of a hyperbola centered at the origin. The hyperbola is the set of all points
  such that the difference of the distances from
  to the foci is constant. See .
If
  is a vertex of the hyperbola, the distance from
  to
  is
  The distance from
  to
  is
  The difference of the distances from the foci to the vertex is
If
  is a point on the hyperbola, we can define the following variables:
By definition of a hyperbola,
  is constant for any point
  on the hyperbola. We know that the difference of these distances is
  for the vertex
  It follows that
  for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula. The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.
This equation defines a hyperbola centered at the origin with vertices
  and co-vertices

### Writing Equations of Hyperbolas in Standard Form

Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.

### Hyperbolas Centered at the Origin

Reviewing the standard forms given for hyperbolas centered at
  we see that the vertices, co-vertices, and foci are related by the equation
  Note that this equation can also be rewritten as
  This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.

### Hyperbolas Not Centered at the Origin

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated
  units horizontally and
  units vertically, the center of the hyperbola will be
  This translation results in the standard form of the equation we saw previously, with
  replaced by
  and
  replaced by

Like hyperbolas centered at the origin, hyperbolas centered at a point
  have vertices, co-vertices, and foci that are related by the equation
  We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.

###  Graphing Hyperbolas Centered at the Origin

When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form
  for horizontal hyperbolas and the standard form
  for vertical hyperbolas.

###  Graphing Hyperbolas Not Centered at the Origin

Graphing hyperbolas centered at a point
  other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms
  for horizontal hyperbolas, and
  for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

### Solving Applied Problems Involving Hyperbolas

As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. See . For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!
The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In  we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

### Key Equations

### Key Concepts

1. A hyperbola is the set of all points
  in a plane such that the difference of the distances between
  and the foci is a positive constant.
2. The standard form of a hyperbola can be used to locate its vertices and foci. See .
3. When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See  and .
4. When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See  and .
5. Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.
For the following exercises, find the equations of the asymptotes for each hyperbola.

### Graphical

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.
For the following exercises, given information about the graph of the hyperbola, find its equation.
For the following exercises, given the graph of the hyperbola, find its equation.

###  Extensions

For the following exercises, express the equation for the hyperbola as two functions, with
  as a function of
  Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.

###  Real-World Applications

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.


# Analytic Geometry

## The Parabola

### Learning Objectives

1. Graph vertical parabolas. (IA 11.2.1)
2. Graph horizontal parabolas. (IA 11.2.2)

### Objective 1: Graph vertical parabolas. (IA 11.2.1)

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here.

### Practice Makes Perfect

Graph vertical parabolas.

### Objective 2: Graph horizontal parabolas. (IA 11.2.2)

Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.

### Practice Makes Perfect

Katherine Johnson is the pioneering NASA mathematician who was integral to the successful and safe flight and return of many human missions as well as satellites. Prior to the work featured in the movie Hidden Figures, she had already made major contributions to the space program. She provided trajectory analysis for the Mercury mission, in which Alan Shepard became the first American to reach space, and she and engineer Ted Sopinski authored a monumental paper regarding placing an object in a precise orbital position and having it return safely to Earth. Many of the orbits she determined were made up of parabolas, and her ability to combine different types of math enabled an unprecedented level of precision.  Johnson said, "You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off."
Johnson's work on parabolic orbits and other complex mathematics resulted in successful orbits, Moon landings, and the development of the Space Shuttle program. Applications of parabolas are also critical to other areas of science. Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.

### Graphing Parabolas with Vertices at the Origin

In The Ellipse, we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola. See .
Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points
       in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
In Quadratic Functions, we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See . Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.
The line segment that passes through the focus and is parallel to the directrix is called the latus rectum. The endpoints of the latus rectum lie on the curve. By definition, the distance
  from the focus to any point
  on the parabola is equal to the distance from
     to the directrix.
To work with parabolas in the coordinate plane, we consider two cases: those with a vertex at the origin and those with a vertex at a point other than the origin. We begin with the former.
Let
     be a point on the parabola with vertex
     focus
     and directrix

as shown in . The distance
  from point
     to point

on the directrix is the difference of the y-values:
     The distance from the focus
  to the point
     is also equal to
     and can be expressed using the distance formula.
Set the two expressions for
     equal to each other and solve for
     to derive the equation of the parabola. We do this because the distance from
     to
     equals the distance from
     to

We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.
The equations of parabolas with vertex
     are
     when the x-axis is the axis of symmetry and
     when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. See . When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.
A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the latus rectum, these lines intersect on the axis of symmetry, as shown in .

###  Writing Equations of Parabolas in Standard Form

In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

### Graphing Parabolas with Vertices Not at the Origin

Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated
       units horizontally and
       units vertically, the vertex will be
       This translation results in the standard form of the equation we saw previously with
       replaced by
       and
       replaced by

To graph parabolas with a vertex
       other than the origin, we use the standard form
       for parabolas that have an axis of symmetry parallel to the x-axis, and
       for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.

### Solving Applied Problems Involving Parabolas

As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry are directed toward any surface of the mirror, the light is reflected directly to the focus. See . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.
Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.

### Key Equations

### Key Concepts

1. A parabola is the set of all points
         in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
2. The standard form of a parabola with vertex
         and the x-axis as its axis of symmetry can be used to graph the parabola. If
         the parabola opens right. If
         the parabola opens left. See .
3. The standard form of a parabola with vertex
         and the y-axis as its axis of symmetry can be used to graph the parabola. If
         the parabola opens up. If
         the parabola opens down. See .
4. When given the focus and directrix of a parabola, we can write its equation in standard form. See .
5. The standard form of a parabola with vertex
         and axis of symmetry parallel to the x-axis can be used to graph the parabola. If
         the parabola opens right. If
         the parabola opens left. See .
6. The standard form of a parabola with vertex
         and axis of symmetry parallel to the y-axis can be used to graph the parabola. If
         the parabola opens up. If
         the parabola opens down. See .
7. Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See .

### Section Exercises

### Verbal

###  Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.
For the following exercises, rewrite the given equation in standard form, and then determine the vertex
         focus
         and directrix
  of the parabola.

### Graphical

For the following exercises, graph the parabola, labeling the focus and the directrix.
For the following exercises, find the equation of the parabola given information about its graph.
For the following exercises, determine the equation for the parabola from its graph.

###  Extensions

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.

###  Real-World Applications


# Analytic Geometry

## Rotation of Axes

### Learning Objectives

1. Using rotation of axes formulas.
2. Identify conic sections by their equations. (IA 11.4.3)

### Objective 1: Using rotation of axes formulas.

If a point

on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle

from the positive
x
-axis, then the coordinates of the point with respect to the new axes are

The following rotations of axes formulas define the relationship between (x,y) and (x’,y’):

### Practice Makes Perfect

Using rotation of axes formulas:

### Objective 2: Identify conic sections by their equations. (IA 11.4.3)

We can identify a conic from its equations by looking at the signs and coefficients of the variables that are squared.

### Practice Makes Perfect

Identify conic sections by their equations.

As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See .
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in . A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.

### Identifying Nondegenerate Conics in General Form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.
where
    and
    are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an
    term that we have not seen in any of the standard form equations. As we will discuss later, the
    term rotates the conic whenever
  is not equal to zero.

### Finding a New Representation of the Given Equation after Rotating through a Given Angle

Until now, we have looked at equations of conic sections without an
     term, which aligns the graphs with the x- and y-axes. When we add an
     term, we are rotating the conic about the origin. If the x-  and y-axes are rotated through an angle, say
     then every point on the plane may be thought of as having two representations:
     on the Cartesian plane with the original x-axis and y-axis, and
  on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See .
We will find the relationships between
     and
     on the Cartesian plane with
     and
     on the new rotated plane. See .
The original coordinate x- and y-axes have unit vectors
     and
     The rotated coordinate axes have unit vectors
     and
     The angle
     is known as the angle of rotation. See . We may write the new unit vectors in terms of the original ones.
Consider a vector in the new coordinate plane. It may be represented in terms of its coordinate axes.
Because
     we have representations of
     and
     in terms of the new coordinate system.

### Writing Equations of Rotated Conics in Standard Form

Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form
    into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the
    and
    coordinate system without the
    term, by rotating the axes by a measure of
    that satisfies
We have learned already that any conic may be represented by the second degree equation
where
    and
    are not all zero. However, if
    then we have an
    term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle
    where

###  Identifying Conics without Rotating Axes

Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is
If we apply the rotation formulas to this equation we get the form
It may be shown that
    The expression does not vary after rotation, so we call the expression invariant. The discriminant,
    is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

### Key Equations

### Key Concepts

1. Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
2. A nondegenerate conic section has the general form
  	   where
  	   and
  	   are not all zero. The values of
  	   and
  	   determine the type of conic. See .
3. Equations of conic sections with an
  	   term have been rotated about the origin. See .
4. The general form can be transformed into an equation in the
  	   and
  	   coordinate system without the
  	   term. See  and .
5. An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine which conic section is represented based on the given equation.
For the following exercises, find a new representation of the given equation after rotating through the given angle.
For the following exercises, determine the angle
     that will eliminate the
     term and write the corresponding equation without the
     term.

### Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.
For the following exercises, graph the equation relative to the
     system in which the equation has no
     term.
For the following exercises, determine the angle of rotation in order to eliminate the
     term. Then graph the new set of axes.
For the following exercises, determine the value of
     based on the given equation.


# 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