# Linear Functions

## Graphs of Linear Functions

Two competing telephone companies offer different payment plans. The two plans charge the same rate per long distance minute, but charge a different monthly flat fee. A consumer wants to determine whether the two plans will ever cost the same amount for a given number of long distance minutes used. The total cost of each payment plan can be represented by a linear function. To solve the problem, we will need to compare the functions. In this section, we will consider methods of comparing functions using graphs.

### Graphing Linear Functions

In Linear Functions, we saw that that the graph of a linear function is a straight line. We were also able to see the points of the function as well as the initial value from a graph. By graphing two functions, then, we can more easily compare their characteristics.
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 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 , we can set  in the equation.
The other characteristic of the linear function is its slope  which is a measure of its steepness. Recall that the slope is the rate of change of the function. The slope of a function is equal to the ratio of the change in outputs to the change in inputs. Another way to think about the slope is by dividing the vertical difference, or rise, by the horizontal difference, or run. We encountered both the y-intercept and the slope in Linear Functions.
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 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 transformations 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

Recall that in Linear Functions, 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. A function may also have an -intercept, 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.
Notice that 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. Notice that 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 y-intercept of the other, they would become the same line.
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 1. So, if
   and
 are negative reciprocals of one another, they can be multiplied together to yield –1. 
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 following equation.
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  We already know that the slope is 3. We just need to determine which value for  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 following function:
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.

### Solving a System of Linear Equations Using a Graph

A system of linear equations includes two or more linear equations. The graphs of two lines will intersect at a single point if they are not parallel. Two parallel lines can also intersect if they are coincident, which means they are the same line and they intersect at every point. For two lines that are not parallel, the single point of intersection will satisfy both equations and therefore represent the solution to the system.
To find this point when the equations are given as functions, we can solve for an input value so that  In other words, we can set the formulas for the lines equal to one another, and solve for the input that satisfies the equation.

### Key Concepts

1. Linear functions may be graphed by plotting points or by using the y-intercept and slope. See  and .
2. Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See .
3. The y-intercept and slope of a line may be used to write the equation of a line.
4. The x-intercept is the point at which the graph of a linear function crosses the x-axis. See  and .
5. Horizontal lines are written in the form,  See .
6. Vertical lines are written in the form,  See .
7. Parallel lines have the same slope.
8. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See .
9. 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 .
10. 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 .
11. A system of linear equations may be solved setting the two equations equal to one another and solving for  The y-value may be found by evaluating either one of the original equations using this x-value.
12. A system of linear equations may also be solved by finding the point of intersection on a graph. See  and .

### Section Exercises


### Verbal


### Algebraic

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 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?

### Graphical

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.
For the following exercises, find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.

### Extensions

For the following exercises, use the functions  

### Real-World Applications
