Vertical Line Test

What is Vertical Line Test?

Functions are well defined if only one output can be assigned to each input. In other words, each x value (input) can have only one y value (output) associated with it. Graphically, this translates into a nice test for functions. The vertical line test states: If the curve graphed is a function, it will pass the vertical line test such that any vertical line drawn on the graph of a function will cross the function in at most one place.

How Do I Use This Activity?

This activity allows users to attempt to build functions: curves that pass the vertical line test.

Controls and Output

  • Get Points- This button creates a new set of five random points on the graph.
  • Reset Board- Clears all the points on the game which allows you to place a new set of points on the graph.
  • No Solution- When it is not possible to create of function from a set of five points click on this. If you have already started creating a function click on retry connecting before clicking on this button.
  • Retry Connecting- If you did not create a function or accidently connected the wrong points you can click on this button to try again.
How to use the Applet
  • Click Get Points to get 5 points on the coordinate system.
  • Try to connect the points to make a function. To connect the points, first click on the one that you want to make the starting point. Then, click on the second, third, fourth, and following points in succession so that a function is created. Note: It is not always possible to build a function from the given points.
  • Click the No Solution button if you believe that the series of points cannot be connected to form a function.
  • The computer will tell you if you build a curve that is not a function -- it will tell you that your graph does not pass the vertical line test.
  • Press reset to roll new points
Scoring
  • This activity will automatically record how successful you are at answering the questions. To view the score, press the Show Score button at the bottom of the activity and a pop-up window will appear with the scoreboard. To close this pop-up window press the Close button or click back on the main window.
  • To pause the scoring, press the Active button at the bottom of the screen and it will change to a Paused button. To resume scoring, press the Paused button.
  • To reset the scoreboard, open the scoreboard using the Show Score button and then press the Reset button.

Description

This activity allows the user to attempt to build curves that pass the vertical line test, so that they are functions. This activity would work well in mixed ability groups of two for about twenty-five minutes if you use the exploration questions and ten minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

  • practice students' function skills
  • practice students' deductive reasoning skills
  • familiarize students with the vertical line test for functions

Standards Addressed

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Functions

  • Interpreting Functions

    • Understand the concept of a function and use function notation
    • Interpret functions that arise in applications in terms of the context
    • Analyze functions using different representations

  • Linear, Quadratic, and Exponential Models

    • Interpret expressions for functions in terms of the situation they model

Grades 9-12

  • Algebra

    • Understand patterns, relations, and functions

Algebra I

  • Algebra

    • Competency Goal 4: The learner will use relations and functions to solve problems.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.

Intermediate Algebra

  • Algebra

    • The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

Algebra I

  • Foundation for Functions

    • 2. The student uses the properties and attributes of functions.

8th Grade

  • Patterns, Functions, and Algebra

    • 8.14a The student will describe and represent relations and functions, using tables, graphs, and rules; and
    • 8.14 The student will

Secondary

  • Algebra II

    • AII.08 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.
    • AII.09 The student will find the domain, range, zeros, and inverse of a function; the value of a function for a given element in its domain; and the composition of multiple functions. Functions will include exponential, logarithmic, and those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions.
    • AII.8
    • AII.9

Textbooks Aligned

8th

  • Module 6 - Visualizing Change

    • Section 1: Graphs and Functions
      • Reason for Alignment: This lesson would be used to supplement the text as the text doesn't use the idea of the vertical line test for understanding if x is a function of y. However the lesson helps sutdents to understand the function concept so would still likely work well here.

Be Prepared to

  • give implicit directions on what they are to do. For example, "Today we are going to explore theworld of functions ... we are going to make lines and test them... then we will try to answer theexploration questions..."
  • answer the question "Why does the vertical line test work"?
  • discuss functions, inputs, and outputs