Subject:
Mathematics, Algebra, Functions
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
• College Algebra
• Functions
• Mathematics
• Tmm0012
Language:
English
Media Formats:
eBook, Interactive, Video

# Functions ## Overview

This material covers Chapter 3: Functions chapter of the OpenStax College Algebra Text. This module contains an overview of learning objectives mapped to the OTM state standards, worksheets that correspond to chapter sections, interactive Desmos Activities that pair with the chapter, and a list of supplemental videos that correspond to the chapter content.

# Chapter 3: Introduction and Outcomes

Functions

A relation associates two elements.  For example we might relate time with distance where we would discuss how far something travels in a given amount of time.  A function is a special type of relation where each independent value is matched with only one dependent value.  In our example, time would be the independent value and distance would be the dependent value because you can determine how far you might travel given an amount of time.  The independent value is our input (domain) and the dependent value is our output (range).

In this chapter, functions will be examined along with their properties.  A library of functions will be introduced and will be used to create piecewise functions and to transform functions.  The graphs of functions will be discussed along with the associated properties.  Functions will be combined using algebraic operations and compositions.  The concept of one-to-one functions will be used to determine if a function has an inverse and then where applicable, the inverse function will be found.

Chapter Sections:

• 3.1: Functions and Function Notation
• 3.2: Domain and Range
• 3.3: Rates of Change and Behavior of Graphs
• 3.4 Composition of Functions
• 3.5: Transformation of Functions
• 3.6: Absolute Value Functions
• 3.7: Inverse Functions

OTM Outcomes

1.       Functions: Successful College Algebra students demonstrate a deep understanding of functions whether they are described verbally, numerically, graphically, or algebraically (both explicitly and implicitly). Students should be proficient working with the following families of functions: linear, quadratic, higher-order polynomial, rational, exponential, logarithmic, radical, and piecewise-defined functions (including absolute value).

1a.  Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.). In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).

1b.  Convert between different representations of a function.

1c.  Perform operations with functions including addition, subtraction, multiplication, division, composition, and inversion; connect properties of constituent functions to properties of the resultant function; and resolve a function into a sum, difference, product, quotient, and/or composite of functions.

4.        Modeling with Functions: Successful College Algebra students should have experience in using and creating mathematics which model a wide range of phenomena.

4a.  Create linear models from data and interpret slope as a rate of change.

Chapter Section Objectives and related OTM standards:

3.1: Functions and Function Notation

• Determine whether a relation represents a function. (1a)
• Find the value of a function. (1c)
• Determine whether a function is one-to-one. (1c)
• Use the vertical line test to identify functions. (1a)
• Graph the functions listed in the library of functions (1a)

3.2: Domain and Range

• Find the domain of a function defined by an equation. (1a)
• Graph piecewise-defined functions. (1a, 1b)

3.3: Rates of Change and Behavior of Graphs

• Find the average rate of change of a function. (4b)
• Use a graph to determine where a function is increasing, decreasing, or constant. (1a)
• Use a graph to locate local maxima and local minima. (1a)
• Use a graph to locate the absolute maximum and absolute minimum. (1a)

3.4 Composition of Functions

• Combine functions using algebraic operations. (1c)
• Create a new function by composition of functions. (1c)
• Evaluate composite functions. (1c)
• Find the domain of a composite function. (1c)
• Decompose a composite function into its component functions. (1c)

3.5: Transformation of Functions

• Graph functions using vertical and horizontal shifts. (1b)
• Graph functions using reflections about the x-axis and the y-axis. (1b)
• Determine whether a function is even, odd, or neither from its graph. (1b)
• Graph functions using compressions and stretches. (1b)
• Combine transformations. (1b)

3.6: Absolute Value Functions

• Graph an absolute value function. (1a)
• Solve an absolute value equation. (N/A)

3.7: Inverse Functions

• Verify inverse functions. (1c)
• Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one. (1c)
• Find or evaluate the inverse of a function. (1c)
• Use the graph of a one-to-one function to graph its inverse function on the same axes. (1c)

# Chapter 3 Worksheets

Please note that seciton 3.6 has been omitted as that content does not fall within the purview of the OTM standards for College Algebra.

The files below are worksheets created to support practice and learning for chatper 3 of the OpenStax College Algebra text.

# Chapter 3 Desmos Activities

The links below will direct to pre-built Desmos activities. These can be easily copied, remixed, or compiled into activities for teachers to customize to their class and objectives. For more on how educators can integrate the Desmos activities into their class please click here to link to the Desmos teacher support site:

Below is a short explanation of the Desmos activates linked.

# Functions Bundle:

The functions bundle contains a large collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Graphing Stories

This activity will help students make the transition from one-variable representations (e.g. number lines) to the TWO-variable representation of the coordinate plane. Students will watch 15-second videos and translate them into graphs with your help.

Function Carnival, Part 2

Using a few contextual situations, this activity develops an understanding of function notation, and how it related to the graph of a function. Most examples are graphing height over time.

Commuting Times

This activity illustrates the relationship between a dataset (which is usually not a function) and a model of the data (which—in algebra—is a function).

Card Sort: Functions

In this activity, students sort graphs, equations, and contexts according to whether each one represents a function.

Marbleslides: Lines

In this delightful and challenging activity, students will transform lines so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Free-Range Functions

This activity challenges students to strengthen their ideas about range. Many students who can correctly indicate the range of (say) y=x^2 using formal mathematical notation may still cling to some mistaken ideas. For example, students commonly say that π is not in the range of y=x^2 because they cannot think of a number that—when squared—gives π.

Polygraph: Twelve Functions

This Custom Polygraph is designed to spark vocabulary-rich conversations about various functions. Key vocabulary that may appear in student questions includes: linear, quadratic, exponential, cubic, absolute value, rational, radical, sinusoid, and step. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language.

Polygraph: Functions and Relations

This Custom Polygraph is designed to spark vocabulary-rich conversations about discrete and continuous functions and relations. Key vocabulary that may appear in student questions includes: function, non-function, relation, discrete, continuous, input, output, x-value, and y-value. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language.

Polygraph: Parent Functions

This Custom Polygraph is designed to spark vocabulary-rich conversations about graphs of parent functions. Key vocabulary that may appear in student questions includes: increasing, decreasing, linear, quadratic, cubic, absolute value, exponential, logarithmic, rational, radical, axis, intercept, and coordinate. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language.

Polygraph: Piecewise Functions

This Custom Polygraph is designed to spark vocabulary-rich conversations about piecewise functions. Key vocabulary that may appear in student questions includes: piecewise, continuous, and interval.

Polygraph: Parabolas Part 2

This activity follows up on Polygraph: Parabolas, using the discussions (and students' informal language) in that activity to develop academic vocabulary related to the graphs of quadratic functions.

Polygraph: World of Functions

This Custom Polygraph is designed to spark vocabulary-rich conversations about various functions. Key vocabulary that may appear in student questions includes: linear, quadratic, exponential, cubic, absolute value, and sinusoid.

# Functions Transformations Bundle:

The transformations functions bundle contains a large collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Polygraph: Transformations

This Custom Polygraph is designed to spark vocabulary-rich conversations about transformation. Key vocabulary that may appear in student questions includes: translation, rotation, reflection, dilation, scale factor, pre-image, and image.

What's My Transformation

In this activity, students explore the idea that all lines are related to each other, as are all parabolas. They extend this idea to a new function type, and they manipulate it to gain skill with symbolic representations of function transformations.

Card Sort: Transformations

This activity asks students to match transformations of graphs to expressions using function notation that generates these transformations.

Marbleslides: Parabolas

In this delightful and challenging activity, students will transform parabolas so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Transformation Golf: Rigid Motion

In this activity, students use their existing understanding of translations, reflections, and rotations to complete a round of transformation golf. For each challenge, their task is the same: Use one or more transformations to transform the pre-image onto the image. We recommend you solve the challenges yourself before assigning this activity.

Symmetry

In this activity, students develop an informal understanding of symmetry of functions. By the end, students should be able to identify the symmetry of a function (reflectional vs rotational) by considering its graph.

Blue Point Rule

Students will observe a red point transform into a blue point by way of a mystery transformation. Students will first write about that transformation verbally, developing their intuition about the transformation, before then writing it algebraically.

Function Transformations: Practice with Symbols

Students will practice describing function transformations using words as well as algebraic notation.

Polygraph: Square Root Functions

This Custom Polygraph is designed to spark vocabulary-rich conversations about square root functions. Key vocabulary that may appear in student questions includes: intercept and quadrant.

Transforming Functions

In this activity, students practice representing graphs of transformations using algebraic notation. The activity gives students timely feedback on their work, letting them see the effect of their algebraic transformations on the graph itself.

Polygraph: Absolute Value

This Custom Polygraph is designed to spark vocabulary-rich conversations about transformations of the absolute value parent function. Key vocabulary that may appear in student questions includes: translation, shift, slide, dilation, stretch, horizontal, vertical, and reflect.

# Chapter 3 Supplemental Videos

Below is a document that links to supplemental videos for this chapter. Note that these videos are not created by the publisher of the text, so some verbiage or problem solving strategies may vary from what is presented in the text.