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.