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 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.

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.

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

Algebra 1

Algebra

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

Algebra I

Algebra

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

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.

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.

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: 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 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

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?

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.

Teacher Input

Lead a class
discussion on the vertical line test.

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.

Independent Practice

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

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.