# Functions

## Inverse Functions

### Learning Objectives

1. Find and evaluate composite functions (IA 10.1.1).
2. Determine whether a function is one-to-one (IA 10.1.2).

### Objective 1: Find and evaluate composite functions (IA 10.1.1).

A composite function is a two-step function and can have numerical or variable inputs.
  is read as “f of g of x”.
To evaluate a composite function, we always start by evaluating the inner function and then evaluate the outer function in terms of the inner function.

### Practice Makes Perfect

Find and evaluate composite functions.

For each of the following function pairs find:

### Objective 2: Determine whether a function is one-to-one (IA 10.1.2).

In creating a process called a function, f(x), it is often useful to undo this process, or create an inverse to the function, f-1(x). When finding the inverse, we restrict our work to one-to-one functions, this means that the inverse we find should also be one-to-one. Remember that the horizontal line test is a great way to check to see if a graph represents a one-to-one function.

For any one-to-one function f(x), the inverse is a function f-1(x) such that  and  .
The following key terms will be important to our understanding of functions and their inverses.
Function: a relation in which each input value yields a unique output value.
Vertical line test: a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once.
One-to-one function: a function for which each value of the output is associated with a unique input value.
Horizontal line test: a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once.

### Practice Makes Perfect

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.
If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards.  provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

###  Verifying That Two Functions Are Inverse Functions

Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula
and substitutes 75 for
   to calculate
Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, Betty gets the week’s weather forecast from  for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.
At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for
   after substituting a value for
   For example, to convert 26 degrees Celsius, she could write
After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.
The formula for which Betty is searching corresponds to the idea of an inverse function, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.
Given a function
   we represent its inverse as
   read as
   inverse of
   The raised
   is part of the notation. It is not an exponent; it does not imply a power of
   . In other words,
   does not mean
   because
   is the reciprocal of
   and not the inverse.
The “exponent-like” notation comes from an analogy between function composition and multiplication: just as
   (1 is the identity element for multiplication) for any nonzero number
   so
   equals the identity function, that is,
This holds for all
   in the domain of
   Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal, some functions do not have inverses.
Given a function
   we can verify whether some other function
   is the inverse of
   by checking whether either
   or
   is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)
For example,
   and
   are inverse functions.
and
A few coordinate pairs from the graph of the function
   are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function
   are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

### Finding Domain and Range of Inverse Functions

The outputs of the function
   are the inputs to
   so the range of
   is also the domain of
   Likewise, because the inputs to
   are the outputs of
   the domain of
   is the range of
   We can visualize the situation as in .
When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of
   is
   because a square “undoes” a square root; but the square is only the inverse of the square root on the domain
   since that is the range of

We can look at this problem from the other side, starting with the square (toolkit quadratic) function
   If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.
In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function
   with its domain limited to
   which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).
If
   on
   then the inverse function is

1. The domain of
   = range of
   =

2. The domain of
   = range of
   =

###  Finding and Evaluating Inverse Functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

### Inverting Tabular Functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.
Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

### Evaluating the Inverse of a Function, Given a Graph of the Original Function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

### Finding Inverses of Functions Represented by Formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example,
   as a function of
    we can often find the inverse function by solving to obtain
    as a function of

### Finding Inverse Functions and Their Graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function
   restricted to the domain
   on which this function is one-to-one, and graph it as in .
Restricting the domain to
   makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.
We already know that the inverse of the toolkit quadratic function is the square root function, that is,
   What happens if we graph both
   and
   on the same set of axes, using the
   axis for the input to both

We notice a distinct relationship: The graph of
   is the graph of
   reflected about the diagonal line
   which we will call the identity line, shown in .
This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

### Key Concepts

1. If
   is the inverse of
   then
   See , , and .
2. Only some of the toolkit functions have an inverse. See .
3. For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
4. A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
5. For a tabular function, exchange the input and output rows to obtain the inverse. See .
6. The inverse of a function can be determined at specific points on its graph. See .
7. To find the inverse of a formula, solve the equation
   for
   as a function of
   Then exchange the labels
   and
   See , , and .
8. The graph of an inverse function is the reflection of the graph of the original function across the line
   See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find
   for each function.
For the following exercises, find a domain on which each function
   is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
   restricted to that domain.
For the following exercises, use function composition to verify that
   and
   are inverse functions.

### Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.
For the following exercises, determine whether the graph represents a one-to-one function.
For the following exercises, use the graph of
   shown in .
For the following exercises, use the graph of the one-to-one function shown in .

### Numeric

For the following exercises, evaluate or solve, assuming that the function
   is one-to-one.
For the following exercises, use the values listed in  to evaluate or solve.

### Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

### Real-World Applications

### Chapter Review Exercises

### Functions and Function Notation

For the following exercises, determine whether the relation is a function.
For the following exercises, evaluate

For the following exercises, determine whether the functions are one-to-one.
For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.
For the following exercises, graph the functions.
For the following exercises, use  to approximate the values.
For the following exercises, use the function
   to find the values in simplest form.

### Domain and Range

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

### Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

### Composition of Functions

For the following exercises, find
   and
    for each pair of functions.
For the following exercises, find
   and the domain for
   for each pair of functions.
For the following exercises, express each function
   as a composition of two functions
   and
   where

### Transformation of Functions

For the following exercises, sketch a graph of the given function.
For the following exercises, sketch the graph of the function
   if the graph of the function
   is shown in .
For the following exercises, write the equation for the standard function represented by each of the graphs below.
For the following exercises, determine whether each function below is even, odd, or neither.
For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

### Absolute Value Functions

For the following exercises, write an equation for the transformation of

For the following exercises, graph the absolute value function.

### Inverse Functions

For the following exercises, find
    for each function.
For the following exercise, find a domain on which the function
    is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
    restricted to that domain.
For the following exercises, use a graphing utility to determine whether each function is one-to-one.

### Practice Test

For the following exercises, determine whether each of the following relations is a function.
For the following exercises, evaluate the function

  at the given input.
For the following exercises, use the functions
   to find the composite functions.
For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.
For the following exercises, determine whether the functions are even, odd, or neither.
For the following exercises, find the inverse of the function.
For the following exercises, use the graph of
   shown in .
For the following exercises, use the graph of the piecewise function shown in .
For the following exercises, use the values listed in .


# Linear Functions

## Introduction to Linear Functions

Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/ In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.
Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data.


# Linear Functions

## Linear Functions

### Learning Objectives

1. Find the slope of a line (IA 3.2.1)
2. Find an equation of the line given two points (IA 3.3.3)

### Objective 1: Find the slope of a line. (IA 3.2.1)

Linear functions are a specific type of function that can be used to model many real-world applications, such as the growth of a plant, earned salary, the distance a train travels over time, or the costs to start a new business. In this section, we will explore linear functions, their graphs, and how to find them using data points.

### Linear Function

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

where  is the initial or starting value of the function (when input, ), and  is the constant rate of change, or slope of the function. The  -intercept is at (  ).

When interpreting slope, it will be important to consider the units of measurement. Make sure to always attach these units to both the numerator and denominator when they are provided to you.

### Practice Makes Perfect

Find the slope of the line.

### Objective 2: Find an equation of the line given two points. (IA 3.3.3)

### Find an Equation of the Line Given Two Points

When data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when two points are given by following the steps below.

### Practice Makes Perfect

Find an equation of the line given two points.
Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train (). It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minuteshttp://www.chinahighlights.com/shanghai/transportation/maglev-train.htm.
Suppose a maglev train travels a long distance, and maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.

### Representing Linear Functions

The function describing the train’s motion is a linear function, which is defined as a function with a constant rate of change. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train’s motion as a function using each method.

### Representing a Linear Function in Word Form

Let’s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.
1. The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.
The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.

### Representing a Linear Function in Function Notation

Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the slope-intercept form of a line, where
         is the input value,
         is the rate of change, and
         is the initial value of the dependent variable.
In the example of the train, we might use the notation
         where the total distance
         is a function of the time
  The rate,
         is 83 meters per second. The initial value of the dependent variable
         is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train.

### Representing a Linear Function in Tabular Form

A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in . From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.

### Representing a Linear Function in Graphical Form

Another way to represent linear functions is visually, using a graph. We can use the function relationship from above,
  to draw a graph as represented in . Notice the graph is a line. When we plot a linear function, the graph is always a line.
The rate of change, which is constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or , of the line. We can see from the graph that the y-intercept in the train example we just saw is
  and represents the distance of the train from the station when it began moving at a constant speed.
Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line

Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.

### Determining Whether a Linear Function Is Increasing, Decreasing, or Constant

The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in (a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in (b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in (c).

### Interpreting Slope as a Rate of Change

In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input,
  and
  and two corresponding values for the output,
  and
  —which can be represented by a set of points,
  and
  —we can calculate the slope

Note that in function notation we can obtain two corresponding values for the output
  and
  for the function

  and
  so we could equivalently write
 indicates how the slope of the line between the points,
  and
  is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.

### Writing and Interpreting an Equation for a Linear Function

Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form and the point-slope form. Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function
  in .
We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose
  and

Now we can substitute the slope and the coordinates of one of the points into the point-slope form.
If we want to rewrite the equation in the slope-intercept form, we would find
If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7. Therefore,
  We now have the initial value
  and the slope
  so we can substitute
  and
  into the slope-intercept form of a line.
So the function is
  and the linear equation would be

### Modeling Real-World Problems with Linear Functions

In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.

### Graphing Linear Functions

Now that we’ve seen and interpreted graphs of linear functions, let’s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-intercept and slope. And the third method is by using transformations of the identity function

### Graphing a Function by Plotting Points

To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function,
  we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point
  Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point
  Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.

### Graphing a Function Using y-intercept and Slope

Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y-intercept, which is the point at which the input value is zero. To find the y-intercept, we can set
  in the equation.
The other characteristic of the linear function is its slope.
Let’s consider the following function.
The slope is
  Because the slope is positive, we know the graph will slant upward from left to right. The y-intercept is the point on the graph when
  The graph crosses the y-axis at
  Now we know the slope and the y-intercept. We can begin graphing by plotting the point
  We know that the slope is the change in the y-coordinate over the change in the x-coordinate. This is commonly referred to as rise over run,
  From our example, we have
  which means that the rise is 1 and the run is 2. So starting from our y-intercept
  we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in .

### Graphing a Function Using Transformations

Another option for graphing is to use a transformation of the identity function
  A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.

### Vertical Stretch or Compression

In the equation
  the
  is acting as the vertical stretch or compression of the identity function. When
  is negative, there is also a vertical reflection of the graph.  Notice in  that multiplying the equation of
  by
  stretches the graph of
  by a factor of
  units if
  and compresses the graph of
  by a factor of
  units if
  This means the larger the absolute value of
  the steeper the slope.

### Vertical Shift

In
  the
  acts as the vertical shift, moving the graph up and down without affecting the slope of the line. Notice in  that adding a value of
  to the equation of
  shifts the graph of
  a total of
  units up if
  is positive and
  units down if
  is negative.
Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.

### Writing the Equation for a Function from the Graph of a Line

Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at . We can see right away that the graph crosses the y-axis at the point
  so this is the y-intercept.
Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point
  To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be
Substituting the slope and y-intercept into the slope-intercept form of a line gives

### Finding the x-intercept of a Line

So far we have been finding the y-intercepts of a function: the point at which the graph of the function crosses the y-axis. Recall that a function may also have an , which is the x-coordinate of the point where the graph of the function crosses the x-axis. In other words, it is the input value when the output value is zero.
To find the x-intercept, set a function
  equal to zero and solve for the value of
  For example, consider the function shown.
Set the function equal to 0 and solve for

The graph of the function crosses the x-axis at the point

### Describing Horizontal and Vertical Lines

There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output, or y-value. In , we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use
  in the equation
  the equation simplifies to
  In other words, the value of the function is a constant. This graph represents the function

A vertical line indicates a constant input, or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.
A vertical line, such as the one in , has an x-intercept, but no y-intercept unless it’s the line
  This graph represents the line

### Determining Whether Lines are Parallel or Perpendicular

The two lines in  are parallel lines: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the other, they would become coincident.
We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.
Unlike parallel lines, perpendicular lines do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in  are perpendicular.
Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is
  So, if
  are negative reciprocals of one another, they can be multiplied together to yield

To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is
  and the reciprocal of
  is 8. To find the negative reciprocal, first find the reciprocal and then change the sign.
As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.
The product of the slopes is –1.

### Writing the Equation of a Line Parallel or Perpendicular to a Given Line

If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.

### Writing Equations of Parallel Lines

Suppose for example, we are given the equation shown.
We know that the slope of the line formed by the function is 3. We also know that the y-intercept is
  Any other line with a slope of 3 will be parallel to
   So the lines formed by all of the following functions will be parallel to

Suppose then we want to write the equation of a line that is parallel to
   and passes through the point
  This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of
   will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.
So
   is parallel to
   and passes through the point

### Writing Equations of Perpendicular Lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.
The slope of the line is 2, and its negative reciprocal is
  Any function with a slope of
  will be perpendicular to
  So the lines formed by all of the following functions will be perpendicular to

As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to
  and passes through the point
  We already know that the slope is
  Now we can use the point to find the y-intercept by substituting the given values into the slope-intercept form of a line and solving for

The equation for the function with a slope of
  and a y-intercept of 2 is
So
  is perpendicular to
  and passes through the point
  Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.

### Key Concepts

1. Linear functions can be represented in words, function notation, tabular form, and graphical form. See .
2. An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. See .
3. Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line. See  and .
4. An equation for a linear function can be written from a graph. See .
5. The equation for a linear function can be written if the slope
  and initial value
  are known. See  and .
6. A linear function can be used to solve real-world problems given information in different forms. See , , and .
7. Linear functions can be graphed by plotting points or by using the y-intercept and slope. See  and .
8. Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See .
9. The equation for a linear function can be written by interpreting the graph. See .
10. The x-intercept is the point at which the graph of a linear function crosses the x-axis. See .
11. Horizontal lines are written in the form,
  See .
12. Vertical lines are written in the form,
  See .
13. Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See .
14. A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x- and y-values of the given point into the equation,
  and using the
  that results. Similarly, the point-slope form of an equation can also be used. See .
15. A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the equation of the curve can be written as a linear function.
For the following exercises, determine whether each function is increasing or decreasing.
For the following exercises, find the slope of the line that passes through the two given points.
For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.
For the following exercises, find the x- and y-intercepts of each equation.
For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
For the following exercises, write an equation for the line described.

### Graphical

For the following exercises, find the slope of the line graphed.
For the following exercises, write an equation for the line graphed.
For the following exercises, match the given linear equation with its graph in .
For the following exercises, sketch a line with the given features.
For the following exercises, sketch the graph of each equation.
For the following exercises, write the equation of the line shown in the graph.

### Numeric

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

### Technology

For the following exercises, use a calculator or graphing technology to complete the task.

### Extensions

For the following exercises, use the functions

### Real-World Applications


# Linear Functions

## Modeling with Linear Functions

### Learning Objectives

1. Graph and interpret applications of slope–intercept form of a linear function. (IA 3.2.5)

### Objective 1: Graph and interpret applications of slope–intercept form of a linear function. (IA 3.2.5)

### Graph and Interpret Applications of Slope–Intercept form of linear equations.

Many real-world applications are modeled by linear functions. We will take a look at a few applications here so you can see how equations written in slope–intercept form describe real world situations.
Usually, when a linear function uses real-world data, different letters are used to represent the variables, instead of using only  and  . The variables used remind us of what quantities are being measured.
Also, we often will need to adjust the axes in our rectangular coordinate system to different scales to accommodate the data in the application. Since many applications have both independent and dependent variables that are positive our graphs will lie primarily in Quadrant I.

### Linear Functions

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

where b is the initial or starting value of the function (when input, x=0), and m is the constant rate of change, or slope of the function. The y-intercept is at (0,b),

  .
When interpreting slope, it will be important to consider the units of measurement. Make sure to always attach these units to both the numerator and denominator when they are provided to you.

### Practice Makes Perfect

Graph and interpret applications of slope–intercept form of a linear function.

Elan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for their trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent this situation? What would be the x-intercept, and what can Elan learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.

### Building Linear Models from Verbal Descriptions

When building linear models to solve problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:
1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
2. Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
3. Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
5. When needed, write a formula for the function.
6. Solve or evaluate the function using the formula.
7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.
Now let’s take a look at the student in Seattle. In Elan’s situation, there are two changing quantities: time and money. The amount of money they have remaining while on vacation depends on how long they stay. We can use this information to define our variables, including units.
So, the amount of money remaining depends on the number of weeks:
  .
Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.
The rate of change is constant, so we can start with the linear model
  Then we can substitute the intercept and slope provided.
To find the t-intercept (horizontal axis intercept), we set the output to zero, and solve for the input.
The t-intercept (horizontal axis intercept) is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Elan will have no money left after 8.75 weeks.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before Elan saved $3,500, but the scenario discussed poses the question once they saved $3,500 because this is when the trip and subsequent spending starts. It is also likely that this model is not valid after the t-intercept (horizontal axis intercept), unless Elan uses a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is

In this example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

### Using a Given Intercept to Build a Model

Some real-world problems provide the vertical axis intercept, which is the constant or initial value. Once the vertical axis intercept is known, the t-intercept (horizontal axis intercept) can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y-intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.
Now we can set the function equal to 0, and solve for
  to find the x-intercept.
The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan.

###  Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

### Using a Diagram to Build a Model

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

### Modeling a Set of Data with Linear Functions

Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in .

### Key Concepts

1. We can use the same problem strategies that we would use for any type of function.
2. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. See .
3. Draw a diagram, where appropriate. See  and .
4. Check for reasonableness of the answer.
5. Linear models may be built by identifying or calculating the slope and using the y-intercept.

### Section Exercises

### Verbal

### Algebraic

For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.
For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.
For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.
For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.
For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.

### Graphical

For the following exercises, use the graph in , which shows the profit,
  in thousands of dollars, of a company in a given year,
  where
  represents the number of years since 1980.
For the following exercises, use the graph in , which shows the profit,
  in thousands of dollars, of a company in a given year,
  where
  represents the number of years since 1980.

### Numeric

For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in . Assume that the house values are changing linearly.
For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in . Assume that the house values are changing linearly.

### Real-World Applications


# Linear Functions

## Fitting Linear Models to Data

### Learning Objectives

1. Plot points in a rectangular coordinate system (IA 3.1.1).
2. Find an equation of the line given two points (IA 3.3.3).

### Objectives: Plot points in a rectangular coordinate system (IA 3.1.1) and find an equation of the line given two points. (IA 3.3.3)

In this section we will be plotting collections of data points and looking for patterns in these data sets. A scatterplot is a collection of points plotted on the same coordinate system. When trying to fit a function to a data set it is important to note if there is a pattern to the data set and whether that pattern is linear or nonlinear. If the dependent variable increases as the independent variable increases, we call this a positive association. If the dependent variable decreases as the independent variable increases, we call this a negative association.

### Practice Makes Perfect

A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.

### Drawing and Interpreting Scatter Plots

A scatter plot is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a linear model, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions.  shows a sample scatter plot.
Notice this scatter plot does not indicate a linear relationship. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.

### Finding the Line of Best Fit

Once we recognize a need for a linear function to model that data, the natural follow-up question is “what is that linear function?” One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the y-intercept. We can approximate the slope of the line by extending it until we can estimate the

### Recognizing Interpolation or Extrapolation

While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation when we predict a value inside the domain and range of the data. The process of extrapolation is used when we predict a value outside the domain and range of the data.
 compares the two processes for the cricket-chirp data addressed in . We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.
There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when
    that is in 50 years, the model would not apply because we could not account for factors fifty years in the future.

### Finding the Line of Best Fit Using a Graphing Utility

While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data valuesTechnically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.. One such technique is called least squares regression and can be computed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculatorsFor example, http://www.shodor.org/unchem/math/lls/leastsq.html. Least squares regression is one means to determine the line that best fits the data, and here we will refer to this method as linear regression.

### Distinguishing Between Linear and Nonlinear Models

As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the correlation coefficient, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a "diagnostic on" selection to find the correlation coefficient, which mathematicians label as
    The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.
We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of
    and the graph of the data,  shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.

### Fitting a Regression Line to a Set of Data

Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.

### Key Concepts

1. Scatter plots show the relationship between two sets of data. See .
2. Scatter plots may represent linear or non-linear models.
3. The line of best fit may be estimated or calculated, using a calculator or statistical software. See .
4. Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See .
5. The correlation coefficient,
    indicates the degree of linear relationship between data. See .
6. A regression line best fits the data. See .
7. The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?

### Graphical

For the following exercises, match each scatterplot with one of the four specified correlations in  and .
For the following exercises, draw a best-fit line for the plotted data.

### Numeric

### Technology

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

### Extensions

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years:

### Real-World Applications

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years:

For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years:

### Chapter Review Exercises

### Linear Functions

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
For the following exercises, find the x- and y- intercepts of the given equation
For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

### Modeling with Linear Functions

For the following exercises, use the graph in  showing the profit,
    in thousands of dollars, of a company in a given year,
    where
    represents years since 1980.
For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.
For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.
For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in . Assume that the house values are changing linearly.

### Fitting Linear Models to Data

For the following exercises, consider the data in , which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years:

### Chapter Practice Test

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.
For the following exercises, use the graph in , showing the profit,
    in thousands of dollars, of a company in a given year,
    where
    represents years since 1980.

For the following exercises, use , which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:


# Polynomial and Rational Functions

## Introduction to Polynomial and Rational Functions

You don't need to dive very deep to feel the effects of pressure. As a person in their neighborhood pool moves eight, ten, twelve feet down, they often feel pain in their ears as a result of water and air pressure differentials. Pressure plays a much greater role at ocean diving depths.
Scuba and free divers are constantly negotiating the effects of pressure in order to experience enjoyable, safe, and productive dives. Gases in a person's respiratory system and diving apparatus interact according to certain physical properties, which upon discovery and evaluation are collectively known as the gas laws. Some are conceptually simple, such as the inverse relationship regarding pressure and volume, and others are more complex. While their formulas seem more straightforward than many you will encounter in this chapter, the gas laws are generally polynomial expressions.


# Polynomial and Rational Functions

## Quadratic Functions

### Learning Objectives

1. Graph quadratic functions using properties. (IA 9.6.4)

### Objective 1: Graph quadratic functions using properties. (IA 9.6.4)

A quadratic function is a function that can be written in the general form , where a, b, and c are real numbers and a≠0. The graph of quadratic function is called a parabola. Parabolas are symmetric around a line (also called an axis) and have the highest (maximum) or the lowest (minimum) point that is called a vertex.

### Making a Table

We can graph quadratic function  by making a table and plotting points.

### Practice Makes Perfect

Graph quadratic function by making a table and plotting points.
Graphing of quadratic functions is much easier when we know the vertex and the axis of symmetry. The vertex of the graph of the quadratic function  in the form  .
The line or axis of symmetry of the parabola is the vertical line  .

### Practice Makes Perfect

Graphing quadratic functions using a vertex.
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. 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, 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 quadratic 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

### Learning Objectives

1. Determine the degree of polynomials (IA 5.1.1).

2. Simplify expressions using properties of exponents (IA 5.2.1).

### Objective 1: Simplify expressions using the properties of exponents (IA 5.2.1).

An exponential expression is an expression that has exponents (or powers).

### Practice Makes Perfect

### Objective 2: Determine the degree of polynomials (IA 5.1.1).

A term can be a number like -2, a variable like x, or a product of numbers and  variables like  .

A polynomial is an expression with more than one term with no variables in the denominator and no negative exponents. Any exponent on the variables must be whole numbers.
For example  are polynomials.
There are three particular types of polynomials:
A monomial is a one term polynomial like  or 2.
A binomial is a two term polynomial like  .
A trinomial is a three term polynomial like  .
The degree of a polynomial in one variable is the highest exponent that appears on the variable in the polynomial. For example, the polynomial  has only one variable,  . The highest exponent on  is 2.

### Power Functions

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 power function is a function that can be represented in the form  where k  and p  are real numbers, and k  is known as the coefficient.

### Practice Makes Perfect

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial, then find the degree of each polynomial.

Determine whether the following functions are power functions. If they are not, state it and the reason why.

### Graph of Power Functions and End Behavior

ⓐ What are the similarities in the graphs of even power functions?
ⓑ What are the similarities in the graphs of the odd power functions?
ⓒ What are the differences between the graphs of the even power functions and the odd power 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

Before we can understand the bird problem, it will be helpful to understand a different 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.
    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 positive 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
  and
  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
  even, are symmetric about the
  axis. In  we see that odd functions of the form
   odd, 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 power function consisting of the leading term. 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 x-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 x-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

### Learning Objectives

1. Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1)
2. Solve a quadratic equation by factoring (IA 6.5.2)

### Objective 1: Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1).

The following outline provides a good strategy for factoring polynomials.

### Practice Makes Perfect

Recognize and use the appropriate method to factor a polynomial completely.

### Objective 2: Solve a quadratic equation by factoring (IA 6.5.2)

If  , where  and  represent real numbers. What can you say about  and  ?

The Zero Product Property states that if  , then  ,  or  , or both.

We can use this property to solve equations.

### Practice Makes Perfect

Solve

### Practice Makes Perfect

Solve a quadratic equation by factoring.

Use the zero factor property to solve each of the following exercises.

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:
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 x-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 shown.
Notice in  that the behavior of the function at each of the x-intercepts is different.
The x-intercept

      is the solution of equation
         The graph passes directly through the x-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 x-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 x-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 x-axis. For zeros with odd multiplicities, the graphs cross or intersect the x-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 x-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 x-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
  (While the theorem is intuitive, the proof is actually quite complicated and requires higher mathematics.) 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 x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-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.