CCSSM – HS Session # 3FunctionsJanuary 17, 2013

Functions are Fundamental

“Functions are truly fundamental to mathematics. In everyday language we say, “The price of a ticket is a function of where you sit,” or “The fuel needed to launch a rocket is a function of its payload.” In each case, the word function expresses the idea that knowledge of one fact tells us another. In mathematics, the most important functions are those in which knowledge of one number tells us another. If we know the length of the side of a square, its area is determined. If the circumference of a circle is known, its radius is determined.”

From Calculus: Single Variable, Second Edition, Hughes-Hallett, Gleason, et al.

A Function is a Tool

“A mathematician needs functions for the same reason that a builder needs hammers and drills. Tools transform things. So do functions. In fact, mathematicians often refer to them as transformations because of this. But instead of wood and steel, the materials that functions pound away on are numbers and shapes, and, sometimes, even other functions.”

Stephen Strogatz, The Joy of X

Formal Definitions of Functions Function Definition Activity

In small groups, sort definitions Discuss with your group:

What characteristics do the definitions have in common?

To what aspects of the function do all of the definitions draw students’ and teachers’ attention?

This activity and other content in this Powerpoint presentation is from the bookDeveloping Essential Understanding of Functions (9 – 12), NCTM (2010)

The Rule of Four

“Where appropriate, topics should be presented Geometrically Numerically Analytically VerballyFrom Calculus: Single Variable, Second Edition, Hughes-Hallett, Gleason, et al.

Outline for Functions Presentation

Grade 8 Function Introduction HS Algebra I Function Topics

Modeling Tasks Covariation and Rates of Change Sequences as Functions Growing Patterns as Functions Families of Functions

Sample Tasks

HS CCSSM Functions Presentation

HS Algebra II Function Topics Polynomial Functions Rational Functions Radical Functions Exponential Functions Logarithmic Functions as Solutions for

Exponentials Trigonometric Functions

Sample Tasks

Function Introduction in Grade 8 The Grade 8 CCSSM expects that students:

Understand that a function is a rule that assigns to each input exactly one output

Understand that the graph of a function is a set of ordered pairs consisting of an input and its corresponding output

Are able to comfortable move between the different representations in the Rule of Four

Interpret y = mx+b as the equation of a linear function Be able to give examples of functions that are not linear

Function Introduction in Grade 8

The Grade 8 CCSSM expects that students can Construct a function to model a linear relationship. Determine the rate of change and initial value,

given: A description of a the relationship Two (x,y) values (from a table or from a graph)

Interpret the rate of change and the initial value In terms of the situation it models In terms of its graph or table of values

Function Introduction in Grade 8

The Grade 8 CCSSM expects that students can Describe qualitatively the functional

relationship between two quantities by Analyzing a graph: increasing, decreasing,

linear, non-linear Sketch a graph that

Exhibits given qualitative features of a function or has been described verbally

CCSSM HS Functions

The Functions Standards are divided into four broad categories, each with a set of clusters

Interpreting Functions (F-IF) Understand concept of and use notation Interpret functions that arise in terms of

the context Analyze functions using different

representations

CCSSM HS Functions

Building Functions (F-BF) Build a function that models a

relationship between two quantities Build new functions from existing

function

CCSSM HS Functions

Linear, Quadratic & Exponential Models (F-LE) Construct and compare linear, quadratic,

and exponential models and solve problems.

Interpret expressions for functions in terms of the situation they model

CCSSM HS Functions

Trigonometric Functions (F-TF) Extend the domain of trigonometric

functions using the unit circle Model periodic phenomena with

trigonometric functions Prove and apply trigonometric identities

Interpreting Functions in Algebra I Understand the concept of a function and use

function notation Focus on

Linear functions Exponential functions Quadratic functions Arithmetic sequences Geometric sequences

(F-IF.1, 2, 3) Read these standards

Interpreting Functions in Algebra I

Graph functions expressed symbolically and show key features by hand in simple cases, or by graphing technology

Analyze Linear functions (show intercepts) Exponential functions (show intercepts and end

behavior) Quadratic functions (show intercepts and

maxima/minima) Absolute value functions (show intercepts) Step functions Piecewise-defined functions

(F.IF.7a, 7b, 7e, 8a, 8b, 9) Read these Standards

Building Functions in Algebra I Build a function that models a relationship

between two quantities (Read F.BF.1, 2) Linear Exponential Quadratic

Build new functions from existing functions (Read F.BF.3,4a) Linear only for F.BF.4a Quadratic Absolute Value

Models in Algebra I Construct and compare linear, quadratic, and

exponential models and solve problems Read F.LE.1a, 1b, 1c, 2, 3

Interpret expressions for functions in terms of the situation they model Linear Exponential of form f(x) = bx + k Read F.LE.5

Circular Animal Pen

Think about all possible circular animal pens enclosed by a length of fencing. For each length of fencing, there is the corresponding area that the fencing will enclose when it is formed into a circle.

Describe the relationship between the area of a circular animal pen and the length of the fencing that it takes to enclose the pen.

From Developing Essential Understanding of Functions, NCTM (2010)

Circular Animal Pen When we use an equation such as A = L2/(2π)to describe

the relationship, we are implying the the “mapping” definition. Each value of L maps to a value of A by following the rule set forth in the equation.

When we sketch of graph of the function, we view the functions as consisting of ordered pairs of the form

[L, L2/(2π)]. When we create a table, we bridge the mapping and

ordered pair definitions. Each entry in the input column maps to the corresponding entry in the output column AND the two entries in each row form an ordered pair.

Covariation and Rate of Change Using a covariation approach is an important

first step in developing a fuller comprehension of the function concept.

The rate of change of a function describes the covariation between two variables (input and output). In other words, how one variable quantity changes with respect to the other variable.

A function’s rate of change is one of the main characteristics that determine what kinds of real world phenomena the function can model.

Rate of Change Tables are a good way to observe covariation

From a covariation perspective (lens), a function is understood as a juxtaposition of two sequences,each of which is generated independently through a patternof values.In the function at right, what stands outfrom a covariation perspective?

x f(x)

-1 10 11 32 73 134 21

Defining Rate of Change

A rate of change is a rate!

change in output value change in input value When students see only tables where the

change in input value is 1, they may think that the rate of change is only the change in the output value!

Defining Rate of Change Average and instantaneous rates of change

Suppose that it starts to rain at noon. Between 2 PM and 6 PM a total of 12 millimeters of

rain falls. What is the rate of change between noon and 6 PM?What is the rate of change between 2 PM and 6 PM?What is the rate of change between 3 PM and 4 PM?What is the rate of change at 5 PM?

Looking at tables through different perspectives

When you look at the table from a

covariation perspective,

what do you notice?

When you look at the table from a

correspondence or mapping perspective,

what you do notice?

x y-2 4-1 10 01 12 43 94 16

Looking at Covariation Patterns for Clues as to Function Family

Research indicates that students typically use a covariation perspective before attempting to generalize the relationships with a mapping perspective

The pattern of covariation gives students a clue as to what function family the table describes A linear function has a constant rate of change

The ratio of the change in output to the change in input is constant

x f(x) = 3x + 2

-1 -10 21 52 83 11

1111

3333

Looking at Covariation Patterns for Clues as to Function Family

A quadratic function has a linear rate of change As the input changes by one unit, the output changes at a

constant rate This constant rate can be seen by finding the second

differencesx f(x) = 3x2

+ 2-1 50 21 52 143 29

1111

-3 3 915

666

Linear Constant

Rate of Change Exponential functions have a rate of change that is

proportional to the value of the function. Whenever the input is increased by 1 unit*, the output is

multiplied by a constant factor, which is the base.

Rate of change = 2*f(x)

*Note that this constant factor isnot as readily seen when the inputis increased by a value other than 1 unit.

x 2x

-1

2-1 = ½

0

20 = ½ * 2 = 1

1

21 = 1*2 = 2

2

22 = 2*2 = 4

3

23 = 4*2 = 8

½1

2

4

Sequences as FunctionsSequences represent a special type of function whose domain is the counting numbers. Arithmetic sequences have a constant rate of change and

are related to linear functions. Geometric sequences have a variable rate of change and are

related to exponential functions. Growing patterns can have constant or variable rates of

change and are generally related to polynomial functions.

Sequences as Functions

Arithmetic Sequences

A sequence is arithmetic if each term beyond the first term can be obtained from the previous term by adding a constant m (m can be positive, negative, or zero).

5, 9, 13, 17, 21…

1.6, 1.1, 0.6, 0.1, -0.4, -0.9, -1.4…

Recursive Rule: NEXT = NOW + m

What is the Recursive Rule for the above two sequences

Sequences as Functions Arithmetic Sequences

1, 4, 7, 10, 13… and 1, 5, 9, 13, 17

The rule that assigns a sequence value to a specific counting, such as “what is the 100th term in the sequence” is harder to recognize.

Find the formulas in terms of N for the Nth entries of the sequences. (Let N = 1 be the first entry in each series) Finding the formula generally requires identifying what the the “0th” entry in the sequence would be.

What relationship do you see between the recursive rules (NEXT = NOW + m and the formulas (input the term # and output the sequence value)?

Sequences as FunctionsGeometric Sequences Define the recursive rule for each sequence below Find the formula in terms of N for any term in each

sequence

10, 20, 40, 80… and 5, 5/3, 5/9, 5/27…

What relationship do you see between the recursive rules and the formulas?

Problems related to F-IF.1, 2, 3

From Illustrative Mathematics:

Interpreting the Graph (Using function notation) http://www.illustrativemathematics.org/illustrations/636

The Customers (Idea of Function in non-math context)http://www.illustrativemathematics.org/illustrations/624

Families of Functions

Functions are often studied and understood as families, and students should spend time studying functions within a family, varying parameters to understand how the parameters affect the graph of the function and its key features.

Families of Functions

F-IF.7 indicates that the following functions should be in students’ repertoires: Linear and quadratic (show intercepts, maxima,

and minima) Square root, cube root, piecewise (step functions

and absolute value functions) Polynomial functions (identify zeros when suitable

factorizations available and show end behavior

Families of FunctionsF-IF.7 indicates that the following functions should be in students’ repertoires: Exponential and logarithmic functions

Showing intercepts and end behavior Trigonometric functions

Showing period, midline, and amplitude (+) Rational functions

Identifying zeros and asymptotes when suitable factorizations available, showing end behavior

Exponential Family

Task: Match graphs to equations by recognizing

transformations, intercepts, end behavior

http://www.illustrativemathematics.org/illustrations/803

F-IF.A.1 Task Suppose f is a function. If 10=f(−4), give the coordinates of a point on the graph of f. If 6 is a solution of the equation f(w)=1, give a point on the graph of f. Commentary:

This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.

This task is adapted from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.

F-IF.A.1 Parking Lot A parking lot charges $0.50 for each half hour or fraction

thereof, up to a daily maximum of $10.00. Let C(t) be the cost in dollars of parking for t minutes.

Complete the table Sketch a graph of C for 0 ≤ t ≤480. Is C a function of t? Explain your reasoning. Is t a function of C? Explain your reasoning.

t (minutes) C(t) (dollars)

015203575125

F-IF.A.2

Let f(t) be the number of people, in millions, who own cell phones t years after 1990. Explain the meaning of the following statements. f(10)=100.3 f(a)=20 f(20)=b n=f(t)from Illustrative Mathematics

Growing Patterns as Functions

See Handout: Seeing Structure and Developing Rules for Patterns

See Fawn Nguyen’s Patterns Poster See Patterns Poster for Algebra I See Fawn Nguyen’s Visual Patterns Website

Functions in Algebra II

Interpret functions that arise in applications in terms of a context (Modeling) Extend modeling with functions introduced in Algebra I to

include trigonometric functions and periodicity

Task: Interpreting the graphic model of an influenza epidemichttp://www.illustrativemathematics.org/illustrations/637

Functions in Algebra II

Analyze functions using different representations Focus on using key features to guide selection of

appropriate type of model function Extend analysis with function introduced in Algebra I to

include polynomial functions and logarithmic functions (Read F-IF.7c,d(+), e)

Analyzing the Graphs of Power Functionshttp://www.illustrativemathematics.org/illustrations/627

Functions in Algebra II

Analyze functions using different representations Write a function defined by an expression in

different but equivalent forms to reveal and explain different properties of the function Factoring and completing the square (Read

F-IF.8a)http://www.illustrativemathematics.org/illustrations/627

Functions in Algebra II

Analyze functions using different representations

Using properties of exponentsUse the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y=(1.02)t, y=(0.97)t, y=(1.01)12t, y=(1.2)t/10, and classify them as representing exponential growth or decay.

Functions in Algebra II

Analyze functions using different representations

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Functions in Algebra II

Building Functions Continue work started in Algebra I on Building

Functions through adding, subtracting, multiplying, and dividing functions.

Include simple radical, rational, and exponential functions

Interpreting and Building an Exponential Modelhttp://www.illustrativemathematics.org/illustrations/533

Functions in Algebra II

Building Functions Identify the effect of rigid transformations and non-rigid

transformations on the domain and range of a function Emphasize common effect of each transformation

across functions types Find inverse functions

Functions in Algebra II

Problem in which students need to analyze graphs of exponential functions (F-IF.7e) and also understand the effect on the graph of of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k)(F-BF.3)

http://www.illustrativemathematics.org/illustrations/803

Functions in Algebra II

Linear, Quadratic, and Exponential Models Continue work started in Algebra I on

modeling with linear, quadratic, and exponential functions

Introduce logarithms as solutions for exponents.

Bacteria problem http://www.illustrativemathematics.org/illustrations/370

Functions in Algebra II

Trigonometric Functions Understand radian measure of an angle as the length

of the arc on the unit circle subtended by the angle. Explain how the unit circle in the coordinate plane

enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Functions in Algebra II

Trigonometric Functions Model periodic phenomena with

trigonometric functions (Read F.TF.5)Foxes and Rabbits 2

As The Wheel TurnsFairly difficult, but accessible, applied trig problem that would be a good group project:

Functions in Algebra II

Trigonometric Functions (Honors or 4th Year) (+) Understand that restricting a trigonometric

function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. ⋆

Functions in Algebra II

Trigonometric Functions Prove and apply trigonometric identities

Prove the Pythagorean identity sin2(θ)+cos2(θ)=1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

More Functions Tasks

F-IF Pizza Place Promotion

F-BF US Households

F-LE Comparing Exponentials

F-LE Newton's Law of Cooling

PARCC Example

Get this from

http://www.parcconline.org/samples/mathematics/high-school-functions