# Linear Functions

## Linear Functions

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 minutes.http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm
Suppose a maglev train were to travel a long distance, and that the train 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, that 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 form known as 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 

in which 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, 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 in  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, that is, 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).

### Calculating and Interpreting Slope

In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the slope given input and output values. 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 

as follows
where  is the vertical displacement and   is the horizontal displacement. Note in function notation 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. The greater the absolute value of the slope, the steeper the line is.

### Writing the Point-Slope Form of a Linear Equation

Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Here, we will learn another way to write a linear function, the point-slope form.
The point-slope form is derived from the slope formula.
Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. We can move from one form to another using basic algebra. For example, suppose we are given an equation in point-slope form, 
. We can convert it to the slope-intercept form as shown.
Therefore, the same line can be described in slope-intercept form as 



### Writing the Equation of a Line Using a Point and the Slope

The point-slope form is particularly useful if we know one point and the slope of a line. Suppose, for example, we are told that a line has a slope of 2 and passes through the point 

We know that 

and that 

and 
 We can substitute these values into the general point-slope equation.
If we wanted to then rewrite the equation in slope-intercept form, we apply algebraic techniques.
Both equations, 

and 
 describe the same line. See .

### Writing the Equation of a Line Using Two Points 

The point-slope form of an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points 

and 
 We can use the coordinates of the two points to find the slope.
Now we can use the slope we found and the coordinates of one of the points to find the equation for the line. Let use (0, 1) for our point.
As before, we can use algebra to rewrite the equation in the slope-intercept form.
Both equations describe the line shown in .

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

Now that we have written equations for linear functions in both the slope-intercept form and the point-slope form, we can choose which method to use 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 
 We can use these points to calculate the slope.
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 wanted 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.

### Key Equations


### Key Concepts

1. The ordered pairs given by a linear function represent points on a line.
2. Linear functions can be represented in words, function notation, tabular form, and graphical form. See .
3. The rate of change of a linear function is also known as the slope.
4. An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
5. The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point at which the line crosses the y-axis.
6. An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
7. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
8. A constant linear function results in a graph that is a horizontal line.
9. Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant. See .
10. 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 .
11. The slope and initial value can be determined given a graph or any two points on the line.
12. One type of function notation is the slope-intercept form of an equation.
13. The point-slope form is useful for finding a linear equation when given the slope of a line and one point. See .
14. The point-slope form is also convenient for finding a linear equation when given two points through which a line passes. See .
15. The equation for a linear function can be written if the slope  and initial value  are known. See , , and .
16. A linear function can be used to solve real-world problems. See  and .
17. A linear function can be written from tabular form. See .

### 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.

### Graphical

For the following exercises, find the slope of the lines graphed.
For the following exercises, write an equation for the lines graphed.

### 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


### Extensions


### Real-World Applications
