Introduction to Functions

Objectives

• Identify examples and non-examples of functions (construct a concept)
• Identify domain and range of constant, quadratic, absolute value, and square root functions (simple knowledge or algorithmic skill depending upon the specific function)
• Utilize the vertical line test to verify if a relation is a function
• Represent functions using tables, graphs, manipulatives, verbal or algebraic rules (algorithmic skill & inductive reasoning)

Teaching Plan

Introduction to Functions ( 10 minutes)

Learners will develope a basic understanding of a function and explore different ways to represent functions.

(1 activity) Discuss the different examples of functions in everyday life. Students name another relationship in everyday life that could be a function.
(1 activity) Students are presented with an overview of mappings, tables, graphs, and notation as ways to represent functions.
(4 activities) Students are presented with examples of functions as different representations (mappings, ordered pairs, graphs, notation) and strengths and weaknesses of using each representation. Students answer questions about functions exploiting the strengths of each representation.

Definition of a Function Functions (15 minutes)

The definition of a function is presented and explained using different representations. The idea of input and output is introduced. Learners will distinguish between functions and non-functions.

(2 activities) Students are presented with the idea of a function as an input and output machine. The input and output are connected with the x and f(x) in functin notation. Students explain why the output would never change for the same input.
(3 activities) The definition of a function is presented and explained using 6 functions represented as mappings, ordered pairs and graphs. Students identify characheristcs functions, and create representations of functions and non-functions.
(1 activity) Students use the vertical line test to identify graphs of functions.
(1 activity) Students explain different examples of functions and non-functions.

Domain and Range (20 minutes)

The idea of input and output is connected to domain and range. Learners identify given domain and range. Learners explain implied domain and range of groups of functions.

(1 activity) Domain and range are introduced as the input and output of the function machine. Students identify the given domain and range of functions.
(3 activities) Domain and range are explained in terms of 6 different functions as different representations (mapping, ordered pairs, graphs). Students name the given domain and range of the functions.
(5 activities) The implied domain and range of constant, linear, quadratic, absolute value, and square root functions are presented and explained. Students name the domaina dn range of similar functions.

Find the Function (10 minutes)

Student use graphing manipulatives to identify function notation given points on the graph of the function.

(1 activity) Students guess the function from the function machine based on inputs and outputs. Students guess the rule for basic functions from a table of ordered pairs.
(8 activities) Students use graphing manilulatives to guess the function notation for linear, quadratic, absolute value, and square root functions. Students are presented with the standard form of the function families and manilulate the coefficient.

Applications of Functions (20 minutes)

Applications of functions are introduced. Students are guided to discover how functions can help solve problems. These may be ideal as a group activity.

(3 activities) Students explore using a function to determine the volume of a box based on given criteria. The implied domain and range of the problem is discussed based on the physical restrictions.
(2 activities) Students explore using a function to determine cost of constructing a pipeline based on given criteria.
(2 activities) Students explore using the graphs of functions to determine altitude and velocity of the space shuttle during launch.
(1 activity) Students explore using the graphs of functions of data generated from a drag racing model.

Assessment (10 minutes)

Formal assessment includes a quiz at the end of the module. The quiz includes questions on different representations of functions, identifying functions, and finding given and implied domain and range.

Review with the class what a fraction describes, and how a fraction can abbreviate a verbal or graphical description of when something is a part of something larger.
Go through the eModule section Representing Fractions with individual students working on computers or as a group using a computer projection system.

Worksheets

None

Credits

Correlation to Standards

Correlation to PSSM Standards

•  Understand patterns, relations, and functions:
  • understand relations and functions and select, convert flexibly among, and use various representations for them;
  • understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;

Correlation to Utah Standards

• Pre-algebra 2.1.1 Represent relations & functions using tables, graphs, manipulatives, verbal or algebraic rules.
• Elementary Algebra 2.3.4 Identify the domain and range of a relation or function from a graph, equation, table, or set of ordered pairs.
• Intermediate Algebra 2.1.1 Compare and contrast relations and functions.
• Intermediate Algebra 2.1.2 Identify the domain and range of the absolute value, quadratic, radical, sine, and cosine functions.
• Precalculus 2.1.1 Identify the domain, range, and other attributes of families of functions and their inverses, i.e., exponential, polynomial, rational, logarithmic, piece-wise, and trigonometric.