How to Graph Step Functions

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  • 0:04 Step Functions
  • 1:35 Graphing Step Functions
  • 2:59 Another Example
  • 4:19 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll learn the definition of a step function and two of its family members: floor functions and ceiling functions. We'll then look at how to graph step functions using two real world examples.

Step Functions

Have you ever been in a taxi watching the meter, only to have it jump up at the end of the ride? Most taxis charge for each mile or increment of a mile, so the price goes up at the end even if you've only driven a small portion of a mile.

For example, consider a taxi that charges a flat rate of $3.00 plus $1.00 per mile or any increment of a mile. The cost is a function of the distance driven (x miles), and is defined in increments or intervals.

Distance Cost
0<x≤1 $4
1<x≤2 $5
2<x≤3 $6
3<x≤4 $7

We see that if our taxi ride was 2.35 miles, for example, we would get charged $6, even though we didn't go a full three miles. When a function is defined in pieces as this one is, we call it a piecewise function, and we represent it using notation that gives an output for each interval of inputs.


Our taxi function is a special kind of piecewise function called a step function. A step function is a function that increases or decreases in steps from one constant value to the next. Within the step function family, there are floor functions and ceiling functions. A floor function is a step function that includes the lower endpoint of each input interval, but not the higher endpoint. A ceiling function is a step function that includes the higher endpoint of each input interval, but not the lower endpoint. Since our taxi function includes the higher endpoint of each input interval, but not the lower, it is a ceiling function.

Graphing Step Functions

There is a very valid reason for the name of a step function. To observe this reason, let's graph our taxi function. Wait! How do we do that? Let's figure it out!

Graphing a step function is the same as graphing any piecewise function. We simply graph each part of it separately. When dealing with a step function, this results in these steps to graph the function:

  1. Draw a horizontal line segment at each constant output value over the interval of input values that it corresponds to.
  2. Draw a closed circle point (a filled in circle) at the included endpoint on each horizontal line.
  3. Draw an open circle point (a circle not filled in) at the endpoint that is not included on each horizontal line.

Well that doesn't seem so hard! Let's graph our taxi function.

First, we draw our horizontal line segments at each cost for the interval of miles that it corresponds to.


Next, we'll draw in our included endpoints with a closed circle.


Lastly, we draw in the endpoints that are not included using an open circle.


Wow, that really was quite easy, and now we have a graph of the step function representing our taxi's fee structure. Do you see why step functions have their name? The graph looks exactly like a series of steps. Pretty neat!

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