# Polynomial and Rational Functions

## Complex Numbers

The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as
Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

### Expressing Square Roots of Negative Numbers as Multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number
  is defined as the square root of negative 1.
So, using properties of radicals,
We can write the square root of any negative number as a multiple of
  Consider the square root of –25.
We use
  and not

because the principal root of
  is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written
  where
  is the real part and
  is the imaginary part. For example,
  is a complex number. So, too, is

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

### Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs
  where
  represents the coordinate for the horizontal axis and
  represents the coordinate for the vertical axis.
Let’s consider the number
  The real part of the complex number is
  and the imaginary part is
  We plot the ordered pair
  to represent the complex number
  as shown in .

### Adding and Subtracting Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

### Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

### Multiplying a Complex Number by a Real Number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,

### Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get
Because
  we have
To simplify, we combine the real parts, and we combine the imaginary parts.

### Dividing Complex Numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of
  is

Note that complex conjugates have a reciprocal relationship: The complex conjugate of
  is
  and the complex conjugate of
  is
  Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide
  by
  where neither
  nor
  equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Apply the distributive property.
Simplify, remembering that

### Simplifying Powers of i

The powers of
  are cyclic. Let’s look at what happens when we raise
  to increasing powers.
We can see that when we get to the fifth power of
  it is equal to the first power. As we continue to multiply
  by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of

### Key Concepts

1. The square root of any negative number can be written as a multiple of
  See .
2. To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See .
3. Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See .
4. Complex numbers can be multiplied and divided.
5. To multiply complex numbers, distribute just as with polynomials. See , , and .
6. To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See , , and .
7. The powers of
  are cyclic, repeating every fourth one. See .

### Verbal

### Algebraic

For the following exercises, evaluate the algebraic expressions.

### Graphical

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.
For the following exercises, plot the complex numbers on the complex plane.

### Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

### Technology

For the following exercises, use a calculator to help answer the questions.

### Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.


# Polynomial and Rational Functions

## Quadratic Functions

Curved antennas, such as the ones shown in , are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.
In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.

### Recognizing Characteristics of Parabolas

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in .
The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of
  at which

### Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions

The general form of a quadratic function presents the function in the form
where
  and
  are real numbers and
  If
  the parabola opens upward. If
  the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.
The axis of symmetry is defined by
  If we use the quadratic formula,
  to solve
  for the
  intercepts, or zeros, we find the value of
  halfway between them is always
  the equation for the axis of symmetry.
 represents the graph of the quadratic function written in general form as
  In this form,
  and
  Because
  the parabola opens upward. The axis of symmetry is
  This also makes sense because we can see from the graph that the vertical line
  divides the graph in half. The vertex always occurs along the axis of symmetry.  For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance,
  The
  intercepts, those points where the parabola crosses the
  axis, occur at
  and

The standard form of a quadratic function presents the function in the form
where
  is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.
As with the general form, if
  the parabola opens upward and the vertex is a minimum. If
  the parabola opens downward, and the vertex is a maximum.  represents the graph of the quadratic function written in standard form as
  Since
  in this example,
  In this form,
  and
  Because
  the parabola opens downward. The vertex is at

The standard form is useful for determining how the graph is transformed from the graph of
   is the graph of this basic function.
If
  the graph shifts upward, whereas if
  the graph shifts downward. In ,
  so the graph is shifted 4 units upward. If
  the graph shifts toward the right and if
  the graph shifts to the left. In ,
  so the graph is shifted 2 units to the left. The magnitude of
  indicates the stretch of the graph. If
  the point associated with a particular
  value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if
  the point associated with a particular
  value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In ,
  so the graph becomes narrower.
The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.
For the linear terms to be equal, the coefficients must be equal.
This is the axis of symmetry we defined earlier. Setting the constant terms equal:
In practice, though, it is usually easier to remember that k is the output value of the function when the input is
  so

### Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.

### Determining the Maximum and Minimum Values of Quadratic Functions

The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola. We can see the maximum and minimum values in .
There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.

### Finding the x- and y-Intercepts of a Quadratic Function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the
  intercept of a quadratic by evaluating the function at an input of zero, and we find the
  intercepts at locations where the output is zero. Notice in  that the number of
  intercepts can vary depending upon the location of the graph.

### Rewriting Quadratics in Standard Form

In , the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

### Key Equations

### Key Concepts

1. A polynomial function of degree two is called a quadratic function.
2. The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
3. The axis of symmetry is the vertical line passing through the vertex. The zeros, or
  intercepts, are the points at which the parabola crosses the
  axis. The
  intercept is the point at which the parabola crosses the
  axis. See , , and .
4. Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See .
5. The vertex can be found from an equation representing a quadratic function. See .
6. The domain of a quadratic function is all real numbers. The range varies with the function. See .
7. A quadratic function’s minimum or maximum value is given by the
  value of the vertex.
8. The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See  and .
9. Some quadratic equations must be solved by using the quadratic formula. See .
10. The vertex and the intercepts can be identified and interpreted to solve real-world problems. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
For the following exercises, determine the domain and range of the quadratic function.
For the following exercises, solve the equations over the complex numbers.
For the following exercises, use the vertex
  and a point on the graph
  to find the general form of the equation of the quadratic function.

### Graphical

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.
For the following exercises, write the equation for the graphed function.

### Numeric

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

### Technology

For the following exercises, use a calculator to find the answer.

### Extensions

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.
For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

### Real-World Applications


# Polynomial and Rational Functions

## Power Functions and Polynomial Functions

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in .
The population can be estimated using the function
     where
       represents the bird population on the island
       years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.

### Identifying Power Functions

In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)
    As an example, consider functions for area or volume. The function for the area of a circle with radius

      is

    and the function for the volume of a sphere with radius

      is
Both of these are examples of power functions because they consist of a coefficient,
  or
  multiplied by a variable
  raised to a power.

### Identifying End Behavior of Power Functions

 shows the graphs of
  and
  which are all power functions with even, positive integer powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.
To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol
  for positive infinity and
  for negative infinity. When we say that “
  approaches infinity,” which can be symbolically written as
  we are describing a behavior; we are saying that
  is increasing without bound.
With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as
  approaches positive or negative infinity, the
  values increase without bound. In symbolic form, we could write
 shows the graphs of
  which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.
These examples illustrate that functions of the form
  reveal symmetry of one kind or another. First, in  we see that even functions of the form
  are symmetric about the
  axis. In  we see that odd functions of the form
  are symmetric about the origin.
For these odd power functions, as

approaches negative infinity,

decreases without bound. As

approaches positive infinity,

increases without bound. In symbolic form we write
The behavior of the graph of a function as the input values get very small (
  ) and get very large (
  ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.
 shows the end behavior of power functions in the form
  where
  is a non-negative integer depending on the power and the constant.

### Identifying Polynomial Functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius

of the spill depends on the number of weeks

that have passed. This relationship is linear.
We can combine this with the formula for the area

of a circle.
Composing these functions gives a formula for the area in terms of weeks.
Multiplying gives the formula.
This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

### Identifying the Degree and Leading Coefficient of a Polynomial Function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

### Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as

gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree. See .

### Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.
We are also interested in the intercepts. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept
  The x-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x-intercept. See .

### Comparing Smooth and Continuous Graphs

The degree of a polynomial function helps us to determine the number of
  intercepts and the number of turning points. A polynomial function of
  degree is the product of
  factors, so it will have at most
  roots or zeros, or
  intercepts. The graph of the polynomial function of degree
  must have at most
  turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.
A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

### Key Equations

### Key Concepts

1. A power function is a variable base raised to a number power. See .
2. The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
3. The end behavior depends on whether the power is even or odd. See  and .
4. A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See .
5. The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See .
6. The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See  and .
7. A polynomial of degree

will have at most

x-intercepts and at most

turning points. See , , , , and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.
For the following exercises, find the degree and leading coefficient for the given polynomial.
For the following exercises, determine the end behavior of the functions.
For the following exercises, find the intercepts of the functions.

### Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.
For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

### Numeric

For the following exercises, make a table to confirm the end behavior of the function.

### Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

### Extensions

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

### Real-World Applications

For the following exercises, use the written statements to construct a polynomial function that represents the required information.


# Polynomial and Rational Functions

## Graphs of Polynomial Functions

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in .
The revenue can be modeled by the polynomial function
where
     represents the revenue in millions of dollars and
       represents the year, with
       corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.

### Recognizing Characteristics of Graphs of Polynomial Functions

Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous.  shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.

###  Using Factoring to Find Zeros of Polynomial Functions

Recall that if
       is a polynomial function, the values of
         for which
         are called zeros of
         If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.
We can use this method to find
  intercepts because at the
  intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:
1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.

### Identifying Zeros and Their Multiplicities

Graphs behave differently at various
  intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.

Suppose, for example, we graph the function
Notice in  that the behavior of the function at each of the
  intercepts is different.
The
  intercept

       is the solution of equation
         The graph passes directly through the
  intercept at

 The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The
  intercept
       is the repeated solution of equation
         The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.
The factor is repeated, that is, the factor
       appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor,
         has multiplicity 2 because the factor
         occurs twice.
The
  intercept
       is the repeated solution of factor
         The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit function
         We call this a triple zero, or a zero with multiplicity 3.
For zeros with even multiplicities, the graphs touch or are tangent to the
  axis. For zeros with odd multiplicities, the graphs cross or intersect the
  axis. See  for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.
For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the
  axis.
For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the
  axis.

### Determining End Behavior

As we have already learned, the behavior of a graph of a polynomial function of the form
will either ultimately rise or fall as

      increases without bound and will either rise or fall as

      decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.
Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function,

 is an even power function, as

 increases or decreases without bound,

 increases without bound. When the leading term is an odd power function, as

 decreases without bound,

 also decreases without bound; as

 increases without bound,

 also increases without bound. If the leading term is negative, it will change the direction of the end behavior.  summarizes all four cases.

###  Understanding the Relationship between Degree and Turning Points

In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function

 in . The graph has three turning points.
This function

is a 4th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.

### Graphing Polynomial Functions

We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.

### Using the Intermediate Value Theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. Consider a polynomial function

 whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers

 and

 in the domain of
  if

 and
  then the function

 takes on every value between

 and

We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function

 at

 lies above the
  axis and another point at

 lies below the
  axis, there must exist a third point between

 and

 where the graph crosses the
  axis. Call this point

 This means that we are assured there is a solution

 where

In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the
  axis.  shows that there is a zero between

and

###  Writing Formulas for Polynomial Functions

Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an
  intercept where each factor is equal to zero, we can form a function that will pass through a set of
  intercepts by introducing a corresponding set of factors.

### Using Local and Global Extrema

With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. These are also referred to as the absolute maximum and absolute minimum values of the function.

### Key Concepts

1. Polynomial functions of degree 2 or more are smooth, continuous functions. See .
2. To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See , , and .
3. Another way to find the
  intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the
  axis. See .
4. The multiplicity of a zero determines how the graph behaves at the
  intercepts. See .
5. The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
6. The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
7. The end behavior of a polynomial function depends on the leading term.
8. The graph of a polynomial function changes direction at its turning points.
9. A polynomial function of degree

has at most

turning points. See .
10. To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most

turning points. See  and .
11. Graphing a polynomial function helps to estimate local and global extremas. See .
12. The Intermediate Value Theorem tells us that if

have opposite signs, then there exists at least one value

between

and

for which

See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the
  or t-intercepts of the polynomial functions.
For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.
For the following exercises, find the zeros and give the multiplicity of each.

### Graphical

For the following exercises, graph the polynomial functions. Note
  and
  intercepts, multiplicity, and end behavior.
For the following exercises, use the graphs to write the formula for a polynomial function of least degree.
For the following exercises, use the graph to identify zeros and multiplicity.
For the following exercises, use the given information about the polynomial graph to write the equation.

### Technology

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

### Extensions

For the following exercises, use the graphs to write a polynomial function of least degree.

### Real-World Applications

For the following exercises, write the polynomial function that models the given situation.


# Polynomial and Rational Functions

## Dividing Polynomials

The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.National Park Service. "Lincoln Memorial Building Statistics." http://www.nps.gov/linc/historyculture/lincoln-memorial-building-statistics.htm. Accessed 4/3/2014 We can easily find the volume using elementary geometry.
So the volume is 73,800 cubic meters

   Suppose we knew the volume, length, and width. We could divide to find the height.
As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any or all of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial

   The length of the solid is given by

   the width is given by
        To find the height of the solid, we can use polynomial division, which is the focus of this section.

### Using Long Division to Divide Polynomials

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.
Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.
We call this the Division Algorithm and will discuss it more formally after looking at an example.
Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide
         by

    using the long division algorithm, it would look like this:

We have found
or
We can identify the dividend, the divisor, the quotient, and the remainder.
Writing the result in this manner illustrates the Division Algorithm.

### Using Synthetic Division to Divide Polynomials

As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.
To illustrate the process, recall the example at the beginning of the section.
Divide

by

using the long division algorithm.
The final form of the process looked like this:

There is a lot of repetition in the table. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.
Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. The process starts by bringing down the leading coefficient.
We then multiply it by the “divisor” and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is

and the remainder is

The process will be made more clear in .

### Using Polynomial Division to Solve Application Problems

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.

### Key Equations

### Key Concepts

1. Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See  and .
2. The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
3. Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form

See , , and .
4. Polynomial division can be used to solve application problems, including area and volume. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use long division to divide. Specify the quotient and the remainder.
For the following exercises, use synthetic division to find the quotient.
For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

### Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.
For the following exercises, use synthetic division to find the quotient and remainder.

### Technology

For the following exercises, use a calculator with CAS to answer the questions.

### Extensions

For the following exercises, use synthetic division to determine the quotient involving a complex number.

### Real-World Applications

For the following exercises, use the given length and area of a rectangle to express the width algebraically.
For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.
For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.


# Polynomial and Rational Functions

## Zeros of Polynomial Functions

A new bakery offers decorated, multi-tiered cakes for display and cutting at Quinceañera and wedding celebrations, as well as sheet cakes to serve most of the guests.. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.

### Evaluating a Polynomial Using the Remainder Theorem

In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by
    the remainder may be found quickly by evaluating the polynomial function at
     that is,
     Let’s walk through the proof of the theorem.
Recall that the Division Algorithm states that, given a polynomial dividend
    and a non-zero polynomial divisor
     where the degree of
  is less than or equal to the degree of
     there exist unique polynomials
     and
     such that
If the divisor,
    is
     this takes the form
Since the divisor
    is linear, the remainder will be a constant,
     And, if we evaluate this for
     we have
In other words,
    is the remainder obtained by dividing
     by

### Using the Factor Theorem to Solve a Polynomial Equation

The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm tells us
If

   is a zero, then the remainder

   is

   and

   or

Notice, written in this form,
    is a factor of
     We can conclude if

   is a zero of
     then
     is a factor of

Similarly, if

   is a factor of

   then the remainder of the Division Algorithm

   is 0. This tells us that

   is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree

   in the complex number system will have

   zeros. We can use the Factor Theorem to completely factor a polynomial into the product of

   factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

### Using the Rational Zero Theorem to Find Rational Zeros

Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial
Consider a quadratic function with two zeros,

   and

   By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.

Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.
We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.

### Finding the Zeros of Polynomial Functions

The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.

### Using the Fundamental Theorem of Algebra

Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.
Suppose
    is a polynomial function of degree four, and
     The Fundamental Theorem of Algebra states that there is at least one complex solution, call it
     By the Factor Theorem, we can write
     as a product of
     and a polynomial quotient. Since
     is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it
     So we can write the polynomial quotient as a product of
     and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of

### Using the Linear Factorization Theorem to Find Polynomials with Given Zeros

A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree

   will have
     zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into
     factors. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form
     where
     is a complex number.
Let

   be a polynomial function with real coefficients, and suppose

   is a zero of

   Then, by the Factor Theorem,

   is a factor of

   For

   to have real coefficients,

   must also be a factor of

   This is true because any factor other than

   when multiplied by

   will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function

   with real coefficients has a complex zero

   then the complex conjugate

   must also be a zero of
     This is called the Complex Conjugate Theorem.

### Using Descartes’ Rule of Signs

There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes’ Rule of Signs tells us of a relationship between the number of sign changes in
    and the number of positive real zeros. For example, the polynomial function below has one sign change.
This tells us that the function must have 1 positive real zero.
There is a similar relationship between the number of sign changes in
    and the number of negative real zeros.
In this case,
    has 3 sign changes. This tells us that
     could have 3 or 1 negative real zeros.

### Solving Real-World Applications

We have now introduced a variety of tools for solving polynomial equations. Let’s use these tools to solve the bakery problem from the beginning of the section.

### Key Concepts

1. To find
      determine the remainder of the polynomial
       when it is divided by
       See .
2.
      is a zero of

     if and only if

     is a factor of
       See .
3. Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See  and .
4. When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
5. Synthetic division can be used to find the zeros of a polynomial function. See .
6. According to the Fundamental Theorem, every polynomial function has at least one complex zero. See .
7. Every polynomial function with degree greater than 0 has at least one complex zero.
8. Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form
      where

     is a complex number. See .
9. The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
10. The number of negative real zeros of a polynomial function is either the number of sign changes of

     or less than the number of sign changes by an even integer. See .
11. Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.
For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.
For the following exercises, use the Rational Zero Theorem to find all real zeros.
For the following exercises, find all complex solutions (real and non-real).

### Graphical

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

### Numeric

For the following exercises, list all possible rational zeros for the functions.

### Technology

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

### Extensions

For the following exercises, construct a polynomial function of least degree possible using the given information.

### Real-World Applications

For the following exercises, find the dimensions of the box described.
For the following exercises, find the dimensions of the right circular cylinder described.


# Polynomial and Rational Functions

## Rational Functions

Suppose we know that the cost of making a product is dependent on the number of items,
  produced. This is given by the equation
  If we want to know the average cost for producing
  items, we would divide the cost function by the number of items,

The average cost function, which yields the average cost per item for
  items produced, is
Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

### Using Arrow Notation

We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in , and notice some of their features.
Several things are apparent if we examine the graph of

1. On the left branch of the graph, the curve approaches the x-axis

2. As the graph approaches
  from the left, the curve drops, but as we approach zero from the right, the curve rises.
3. Finally, on the right branch of the graph, the curves approaches the x-axis

To summarize, we use arrow notation to show that
  or
  is approaching a particular value. See .

### Local Behavior of

Let’s begin by looking at the reciprocal function,
  We cannot divide by zero, which means the function is undefined at
  so zero is not in the domain. As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in .
We write in arrow notation
As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in .
We write in arrow notation
See .
This behavior creates a vertical asymptote, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line
  as the input becomes close to zero. See .

### End Behavior of

As the values of
  approach infinity, the function values approach 0. As the values of
  approach negative infinity, the function values approach 0. See . Symbolically, using arrow notation

Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line
  See .

### Solving Applied Problems Involving Rational Functions

In , we shifted a toolkit function in a way that resulted in the function
  This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.

### Finding the Domains of Rational Functions

A vertical asymptote represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.

### Identifying Vertical Asymptotes of Rational Functions

By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.

### Vertical Asymptotes

The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.

### Removable Discontinuities

Occasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity.
For example, the function
  may be re-written by factoring the numerator and the denominator.
Notice that
  is a common factor to the numerator and the denominator. The zero of this factor,
  is the location of the removable discontinuity. Notice also that
  is not a factor in both the numerator and denominator. The zero of this factor,
  is the vertical asymptote. See .

### Identifying Horizontal Asymptotes of Rational Functions

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at

In this case, the end behavior is
  This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function
  and the outputs will approach zero, resulting in a horizontal asymptote at
  See . Note that this graph crosses the horizontal asymptote.
Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.
In this case, the end behavior is
  This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function
  As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of
  looks like a diagonal line, and since
  will behave similarly to
  it will approach a line close to
  This line is a slant asymptote.
To find the equation of the slant asymptote, divide
  The quotient is
  and the remainder is 2. The slant asymptote is the graph of the line
  See .
Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at
  where
  and
  are the leading coefficients of
  and
  for

In this case, the end behavior is
  This tells us that as the inputs grow large, this function will behave like the function
  which is a horizontal line. As
  resulting in a horizontal asymptote at
  See . Note that this graph crosses the horizontal asymptote.
Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function
with end behavior
the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.

### Graphing Rational Functions

In , we see that the numerator of a rational function reveals the x-intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have integer powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials.
The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. See .
When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See .
For example, the graph of
  is shown in .
1. At the x-intercept
  corresponding to the
  factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.
2. At the x-intercept
  corresponding to the
  factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
3. At the vertical asymptote
  corresponding to the
  factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function

4. At the vertical asymptote
  corresponding to the
  factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side.

### Writing Rational Functions

Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an x-intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set of x-intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.

### Key Equations

### Key Concepts

1. We can use arrow notation to describe local behavior and end behavior of the toolkit functions
  and
  See .
2. A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See .
3. Application problems involving rates and concentrations often involve rational functions. See .
4. The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See .
5. The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See .
6. A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See .
7. A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See , , , and .
8. Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See .
9. If a rational function has x-intercepts at
  vertical asymptotes at
  and no
  then the function can be written in the form
See

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the domain of the rational functions.
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.
For the following exercises, find the x- and y-intercepts for the functions.
For the following exercises, describe the local and end behavior of the functions.
For the following exercises, find the slant asymptote of the functions.

### Graphical

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.
For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.
For the following exercises, write an equation for a rational function with the given characteristics.
For the following exercises, use the graphs to write an equation for the function.

### Numeric

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

### Technology

For the following exercises, use a calculator to graph
  Use the graph to solve

### Extensions

For the following exercises, identify the removable discontinuity.

### Real-World Applications

For the following exercises, express a rational function that describes the situation.
For the following exercises, use the given rational function to answer the question.
For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.


# Polynomial and Rational Functions

## Inverses and Radical Functions

Park rangers and other trail managers may construct rock piles, stacks, or other arrangements, usually called cairns, to mark trails or other landmarks. (Rangers and environmental scientists discourage hikers from doing the same, in order to avoid confusion and preserve the habitats of plants and animals.) A cairn in the form of a mound of gravel is in the shape of a cone with the height equal to twice the radius.
The volume is found using a formula from elementary geometry.
We have written the volume
  in terms of the radius
  However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula
This function is the inverse of the formula for
  in terms of

In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.

### Finding the Inverse of a Polynomial Function

Two functions
  and
  are inverse functions if for every coordinate pair in
  there exists a corresponding coordinate pair in the inverse function,
  In other words, the coordinate pairs of the inverse functions have the input and output interchanged.
For a function to have an inverse, it must be one-to-one.
For example, suppose the Sustainability Club builds a water runoff collector in the shape of a parabolic trough as shown in . We can use the information in the figure to find the surface area of the water in the trough as a function of the depth of the water.
Because it will be helpful to have an equation for the parabolic cross-sectional shape, we will impose a coordinate system at the cross section, with
  measured horizontally and
  measured vertically, with the origin at the vertex of the parabola. See .
From this we find an equation for the parabolic shape. We placed the origin at the vertex of the parabola, so we know the equation will have form
  Our equation will need to pass through the point (6, 18), from which we can solve for the stretch factor

Our parabolic cross section has the equation
We are interested in the surface area of the water, so we must determine the width at the top of the water as a function of the water depth. For any depth
  the width will be given by
  so we need to solve the equation above for
  and find the inverse function. However, notice that the original function is not one-to-one, and indeed, given any output there are two inputs that produce the same output, one positive and one negative.
To find an inverse, we can restrict our original function to a limited domain on which it is one-to-one. In this case, it makes sense to restrict ourselves to positive
  values. On this domain, we can find an inverse by solving for the input variable:
This is not a function as written. Since we are limiting ourselves to positive
  values in the original function, we can eliminate the negative solution, which gives us the inverse function we’re looking for.
Because
  is the distance from the center of the parabola to either side, the entire width of the water at the top will be
  The trough is 3 feet (36 inches) long, so the surface area will then be:
This example illustrates two important points:
1. When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.
2. The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.
Functions involving roots are often called radical functions. While it is not possible to find an inverse of most polynomial functions, some basic polynomials do have inverses. Such functions are called invertible functions, and we use the notation

Warning:
  is not the same as the reciprocal of the function
  This use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a function
  we would need to write

An important relationship between inverse functions is that they “undo” each other. If
  is the inverse of a function

then
  is the inverse of the function
  In other words, whatever the function
  does to

  undoes it—and vice-versa. More formally, we write
and

### Restricting the Domain to Find the Inverse of a Polynomial Function

So far, we have been able to find the inverse functions of cubic functions without having to restrict their domains. However, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would then have an inverse function. Since quadratic functions are not one-to-one, we must restrict their domain in order to find their inverses.

### Solving Applications of Radical Functions

Notice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we want to find the inverse of a radical function, we will need to restrict the domain of the answer because the range of the original function is limited.

###

Radical functions are common in physical models, as we saw in the section opener. We now have enough tools to be able to solve the problem posed at the start of the section.

### Determining the Domain of a Radical Function Composed with Other Functions

When radical functions are composed with other functions, determining domain can become more complicated.

### Finding Inverses of Rational Functions

As with finding inverses of quadratic functions, it is sometimes desirable to find the inverse of a rational function, particularly of rational functions that are the ratio of linear functions, such as in concentration applications.

### Key Concepts

1. The inverse of a quadratic function is a square root function.
2. If

is the inverse of a function

then

is the inverse of the function

See .
3. While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See .
4. To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See  and .
5. When finding the inverse of a radical function, we need a restriction on the domain of the answer. See  and .
6. Inverse and radical and functions can be used to solve application problems. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the inverse of the function on the given domain.
For the following exercises, find the inverse of the functions.
For the following exercises, find the inverse of the functions.

### Graphical

For the following exercises, find the inverse of the function and graph both the function and its inverse.
For the following exercises, use a graph to help determine the domain of the functions.

### Technology

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

###  Extensions

For the following exercises, find the inverse of the functions with
  positive real numbers.

### Real-World Applications

For the following exercises, determine the function described and then use it to answer the question.


# Polynomial and Rational Functions

## Modeling Using Variation

A pre-owned car dealer has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. As she considers the offer, she takes into account the typical price of the dealer's cars, the overall market, and how many she can reasonably expect to sell. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.

### Solving Direct Variation Problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula
 tells us her earnings,
 come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See .
 Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation. Each variable in this type of relationship varies directly with the other.
 represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula
 is used for direct variation. The value
 is a nonzero constant greater than zero and is called the constant of variation. In this case,
 and

### Solving Inverse Variation Problems

Water temperature in an ocean varies inversely to the water’s depth. Between the depths of 250 feet and 500 feet, the formula
 gives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.
If we create , we observe that, as the depth increases, the water temperature decreases.
We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations.
For our example,  depicts the inverse variation. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula
 for inverse variation in this case uses

### Solving Problems Involving Joint Variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable
  cost, varies jointly with the number of students,
  and the distance,

### Key Equations

### Key Concepts

1. A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See .
2. Two variables that are directly proportional to one another will have a constant ratio.
3. A relationship where one quantity is a constant divided by another quantity is called inverse variation. See .
4. Two variables that are inversely proportional to one another will have a constant multiple. See .
5. In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, write an equation describing the relationship of the given variables.

### Numeric

For the following exercises, use the given information to find the unknown value.

### Technology

For the following exercises, use a calculator to graph the equation implied by the given variation.

###  Extensions

For the following exercises, use Kepler’s Law, which states that the square of the time,
 required for a planet to orbit the Sun varies directly with the cube of the mean distance,
 that the planet is from the Sun.

###  Real-World Applications

For the following exercises, use the given information to answer the questions.

### Chapter Review Exercises

You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.

### Complex Numbers

Perform the indicated operation with complex numbers.
Solve the following equations over the complex number system.

### Quadratic Functions

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.
For the following problems, find the equation of the quadratic function using the given information.
Answer the following questions.

### Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.
For the following exercises, determine end behavior of the polynomial function.

### Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.
For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

### Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.
For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

### Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.
For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

### Rational Functions

For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.
For the following exercises, find the slant asymptote.

### Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

### Modeling Using Variation

For the following exercises, find the unknown value.
For the following exercises, solve the application problem.

### Chapter Test

Perform the indicated operation or solve the equation.
Give the degree and leading coefficient of the following polynomial function.
Determine the end behavior of the polynomial function.
Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.
Given information about the graph of a quadratic function, find its equation.
Solve the following application problem.
Find all zeros of the following polynomial functions, noting multiplicities.
Based on the graph, determine the zeros of the function and multiplicities.
Use long division to find the quotient.
Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.
Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.
Given the following information about a polynomial function, find the function.
Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.
For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.
Find the slant asymptote of the rational function.
Find the inverse of the function.
Find the unknown value.
Solve the following application problem.


# Exponential and Logarithmic Functions

## Introduction to Exponential and Logarithmic Functions

Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.
Bacteria commonly reproduce through a process called binary fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to days or years.Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html.
For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour.  shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to extrapolate the table to twenty-four hours, we would have over 16 million!
In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data.


# Exponential and Logarithmic Functions

## Exponential Functions

### Learning Objectives

1. Find the value of a function (exponential). (IA 3.5.3)
2. Graph exponential functions. (IA 10.2.1)

### Objective 1: Find the value of a function (exponential). (IA 3.5.3)

### Practice Makes Perfect

Find the value of an exponential function.

### Objective 2: Graph exponential functions. (IA 10.2.1)

### Practice Makes Perfect

Graph exponential functions.

The number e, e ≈ 2.718281827, is like the number π in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base or Euler's number after the Swiss mathematician Leonhard Euler.
The exponential function whose base is e,  is called the natural exponential function.

### Practice Makes Perfect

India is the second most populous country in the world with a population of about
 billion people in 2021. The population is growing at a rate of about
 each yearhttp://www.worldometers.info/world-population/. Accessed February 24, 2014.. If this rate continues, the population of India will exceed China’s population by the year
 When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.

### Identifying Exponential Functions

When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. For example, in the equation
 the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.

### Defining an Exponential Function

A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products—no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.
What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.

For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See .
From  we can infer that for these two functions, exponential growth dwarfs linear growth.
1. Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
2. Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.
Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one.  For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.
The general form of the exponential function is
 where
 is any nonzero number,
 is a positive real number not equal to 1.
1. If
 the function grows at a rate proportional to its size.
2. If
 the function decays at a rate proportional to its size.
Let’s look at the function
 from our example. We will create a table () to determine the corresponding outputs over an interval in the domain from

 to

Let us examine the graph of
 by plotting the ordered pairs we observe on the table in , and then make a few observations.
Let’s define the behavior of the graph of the exponential function
 and highlight some its key characteristics.
1. the domain is

2. the range is

3. as

4. as

5.
 is always increasing,
6. the graph of
 will never touch the x-axis because base two raised to any exponent never has the result of zero.
7.
 is the horizontal asymptote.
8. the y-intercept is 1.

### Evaluating Exponential Functions

Recall that the base of an exponential function must be a positive real number other than
 Why do we limit the base
 to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:
1. Let
 and
 Then
 which is not a real number.
Why do we limit the base to positive values other than
 Because base
 results in the constant function. Observe what happens if the base is

1. Let
 Then
 for any value of

To evaluate an exponential function with the form
 we simply substitute
 with the given value, and calculate the resulting power. For example:
Let
 What is

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:
Let
 What is

Note that if the order of operations were not followed, the result would be incorrect:

### Defining Exponential Growth

Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.
In more general terms, we have an exponential function, in which a constant base is raised to a variable exponent. To differentiate between linear and exponential functions, let’s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function
 Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function

A few years of growth for these companies are illustrated in .
The graphs comparing the number of stores for each company over a five-year period are shown in . We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.
Notice that the domain for both functions is
 and the range for both functions is
 After year 1, Company B always has more stores than Company A.
Now we will turn our attention to the function representing the number of stores for Company B,
 In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and
 represents the growth factor. Generalizing further, we can write this function as
 where 100 is the initial value,
 is called the base, and
 is called the exponent.

### Finding Equations of Exponential Functions

In the previous examples, we were given an exponential function, which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, then determine the constants
 and
 and evaluate the function.

### Applying the Compound-Interest Formula

Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time
 principal
 APR
 and number of compounding periods in a year

For example, observe , which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.

### Evaluating Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases.  shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in .
These values appear to be approaching a limit as
 increases without bound. In fact, as
 gets larger and larger, the expression
 approaches a number used so frequently in mathematics that it has its own name: the letter
 This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

### Investigating Continuous Growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use
 as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

### Key Equations

### Key Concepts

1. An exponential function is defined as a function with a positive constant other than
 raised to a variable exponent. See .
2. A function is evaluated by solving at a specific value. See  and .
3. An exponential model can be found when the growth rate and initial value are known. See .
4. An exponential model can be found when the two data points from the model are known. See .
5. An exponential model can be found using two data points from the graph of the model. See .
6. An exponential model can be found using two data points from the graph and a calculator. See .
7. The value of an account at any time
 can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See .
8. The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See .
9. The number
 is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is

10. Scientific and graphing calculators have the key
 or
 for calculating powers of
 See .
11. Continuous growth or decay models are exponential models that use
 as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, identify whether the statement represents an exponential function. Explain.
For the following exercises, consider this scenario: For each year
 the population of a forest of trees is represented by the function
 In a neighboring forest, the population of the same type of tree is represented by the function
 (Round answers to the nearest whole number.)
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
For the following exercises, find the formula for an exponential function that passes through the two points given.
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
For the following exercises, use the compound interest formula,

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

### Numeric

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

### Technology

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

### Extensions

### Real-World Applications