Functions and the Vertical Line Test

Abstract

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.

Objectives

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

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

  • Building Functions

    • Build a function that models a relationship between two quantities
    • Build new functions from existing 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

    • Construct and compare linear, quadratic, and exponential models and solve problems
    • Interpret expressions for functions in terms of the situation they model

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Use mathematical models to represent and understand quantitative relationships

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Introductory Mathematics

  • Algebra

    • COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.
    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Algebra I

  • Algebra

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

Technical Mathematics II

  • Data Analysis and Probability

    • Competency Goal 2: The learner will use relations and functions to solve problems.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.

Intermediate Algebra

  • Algebra

    • The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • The student will demonstrate through the mathematical processes an understanding of functions, systems of equations, and systems of linear inequalities.

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.10 The student will investigate and describe through the use of graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression.
    • AII.19 The student will collect and analyze data to make predictions and solve practical problems. Graphing calculators will be used to investigate scatterplots and to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions.
    • AII.8
    • AII.10
    • AII.19

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.

Student Prerequisites

  • Arithmetic: Students 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 functions

Functions that stay the same no matter what the variable does are called constant functions

constants

In math, things that do not change are called constants. The things that do change are called variables.

continuous graph

In a graph, a continuous line with no breaks in it forms a continuous graph

discontinuous graph

A line in a graph that is interrupted, or has breaks in it forms a discontinuous graph

function

A 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

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

input

The number or value that is entered, for example, into a function machine. The number that goes into the machine is the input

origin

In the Cartesian coordinate plane, the origin is the point at which the horizontal and vertical axes intersect, at zero (0,0)

output

The 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.