Functions and the Vertical Line Test

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The following discussions and activities are designed to lead the students to explore the the vertical line test for functions. Plotting points and drawing simple piecewise functions are practiced along the way.

This lesson can be done with individual students or in groups of any size. It is a brief lesson, with the short version taking as little and 30 minutes.


Upon completion of this lesson, students will:

  • be able to recognize functions from graphs
  • be able to recognize functions as formulas
  • have learned how to use the vertical line test to verify if a curve is a function
  • have practiced their point and function plotting skills

Standards Addressed:

Textbooks Aligned:

Student Prerequisites

  • Arithmetic: Student must be able to:
    • perform integer and fractional arithmetic
    • plot points on the Cartesian coordinate system
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities
  • Algebraic: Students must be able to:
    • work with simple algebraic expressions.

Teacher Preparation

  • Access to a browser
  • Pencil
  • Two dice, preferably of different colors.

Key Terms

constant functionsFunctions that stay the same no matter what the variable does are called constant functions
constantsIn math, things that do not change are called constants. The things that do change are called variables.
continuous graphIn a graph, a continuous line with no breaks in it forms a continuous graph
discontinuous graphA line in a graph that is interrupted, or has breaks in it forms a discontinuous graph
functionA function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important
graphA visual representation of data that displays the relationship among variables, usually cast along x and y axes.
inputThe number or value that is entered, for example, into a function machine. The number that goes into the machine is the input
originIn the Cartesian coordinate plane, the origin is the point at which the horizontal and vertical axes intersect, at zero (0,0)
outputThe number or value that comes out from a process. For example, in a function machine, a number goes in, something is done to it, and the resulting number is the output

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • We have been practicing plotting points on the cartesian coordinate plane. (Draw a line on a graph on the board.) Does anyone have any ideas on how we could tell people how to draw this exact same line on another graph without showing it to them?

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we are going to learn about graphing functions.
    • We are going to use the computers to learn about graphing functions, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

    • Lead a class discussion on the vertical line test.

  4. Guided Practice

    • Practice with the students the Simple Plot exercise so that they can practice plotting ordered pairs.
    • Have the students then practice graphing skills on graph paper using the tables of values they generated in the Functions and Linear Functions lessons, using the vertical line test to verify that the graphs represent functions.

  5. Independent Practice

    • Have the students try the computer version of the Vertical Line Test activity to practice applying the vertical line test.

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways:

  • Do only the vertical line discussion and function checker activity.
  • Add a discussion about fractional movement on the coordinate plane.
  • Limit the exercise to the positive domain only.

Suggested Follow-Up

After these discussions and activities, the students will have a good foundation for simple function, function notation, and the vertical line test.

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