Coordinate Algebra: Unit 3 – Linear and Exponential Functions 3a (4 weeks) & 3b (3 weeks) (7 weeks) Unit Overview: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. This unit lays the foundation for the entire course. The concept of a function is threaded throughout each unit in Coordinate Algebra and acts as a bridge to future courses. Students will develop a critical understanding of the concept of a function by examining linear functions and comparing and contrasting them with exponential functions. Note that exponential functions are restricted to those of the form: f(x) =b x + k, where b > 1, k is an integer and x is any real number. Content Standards: Represent and solve equations and inequalities graphically MCC9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.) MCC9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value , exponential, and logarithmic functions. ★ Understand the concept of a function and use function notation MCC9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Draw examples from linear and exponential functions.) MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Draw examples from linear and exponential functions.) MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.) Interpret functions that arise in applications in terms of the context MCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;
Transcript
Coordinate Algebra: Unit 3 – Linear and Exponential Functions3a (4 weeks) & 3b (3 weeks)
(7 weeks)Unit Overview: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.
This unit lays the foundation for the entire course. The concept of a function is threaded throughout each unit in Coordinate Algebra and acts as a bridge to future courses. Students will develop a critical understanding of the concept of a function by examining linear functions and comparing and contrasting them with exponential functions. Note that exponential functions are restricted to those of the form: f(x) =bx + k, where b > 1, k is an integer and x is any real number.
Content Standards: Represent and solve equations and inequalities graphicallyMCC9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.)MCC9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
Understand the concept of a function and use function notationMCC9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Draw examples from linear and exponential functions.)MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Draw examples from linear and exponential functions.)MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.)
Interpret functions that arise in applications in terms of the contextMCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ (Focus on linear and exponential functions.)MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★ (Focus on linear and exponential functions.)MCC9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.)
Analyze functions using different representationsMCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★ MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.★ MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)
Build a function that models a relationship between two quantitiesMCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★ (Limit to linear and exponential functions.)MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.)MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Limit to linear and exponential functions.)
MCC9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★
Build new functions from existing functionsMCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.)
Construct and compare linear, quadratic , and exponential models and solve problems MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★ MCC9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.★ MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.★ MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.★ MCC9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).★ MCC9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★
Interpret expressions for functions in terms of the situation they modelMCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★ (Limit exponential functions to those of the form f(x) = bx + k.)
Standards for Mathematical Practice: 4 Model with mathematics.8 Look for and express regularity in repeated reasoning.
Coordinate Algebra: Unit 3a – Linear and Exponential Functions(4 weeks)
Content Standards: Represent and solve equations and inequalities graphicallyMCC9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.)MCC9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
Understand the concept of a function and use function notationMCC9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Draw examples from linear and exponential functions.)MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Draw examples from linear and exponential functions.)MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.)
Interpret functions that arise in applications in terms of the contextMCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ (Focus on linear and exponential functions.)MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★ (Focus on linear and exponential functions.)MCC9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.)
Analyze functions using different representationsMCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★ MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.★ MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)
Standards for Mathematical Practice: 4 Model with mathematics.8 Look for and express regularity in repeated reasoning.
Standards for Mathematical Practice (4, 8)
EQ: How do mathematically proficient students use mathematical models to solve problems? (MP4) How can recognizing repetition or regularity help solve problems more efficiently? (MP8)
Learning Targets:I can …
apply the mathematics I know to solve problems arising in everyday life, society, and the workplace. (MP4) write an equation to describe a situation. (MP4)
apply what I know to make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. (MP4)
identify important quantities in a practical situation. (MP4) map quantity relationships using such tools as diagrams, tables, graphs, and formulas. (MP4) analyze relationships mathematically to draw conclusions. (MP4)
interpret my mathematical results in the context of the situation. (MP4) reflect on whether my results make sense, possibly improving the model if it has not served its purpose. (MP4) notice if calculations are repeated, and look both for general methods and for shortcuts. (MP8) maintain oversight of the problem solving process, while also attending to the details. (MP8) continually evaluate the reasonableness of my intermediate results. (MP8)
Concept Overview: MP4 Model with mathematics.Linear and exponential functions often serve as effective models for real life contexts. Teachers who are developing students’ capacity to "model with mathematics" move explicitly between real-world scenarios and mathematical representations of those scenarios. Teachers might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted. One strategy for developing this skill is to pose scenarios with no question, and ask student to complete the statements, “I notice …, I wonder…” For sample scenarios, click here.
MP8 Look for and express regularity in repeated reasoning.In this unit, students will have the opportunity to explore linear and exponential functions using tables. In the Make a Table strategy (which should really be called Make a Table and Look for Patterns) students have the opportunity to explore and talk through patterns they see in repeated calculations. Students are encouraged to look for and describe patterns both horizontally and vertically, as well as to describe what’s happening “over and over again.” Even the simple activity provided in an extension of the Guess and Check activities (courtesy of Dr. Math), in which students do calculations without ever simplifying any steps, help them see and exploit repeated reasoning.
Resources:MP4 Inside Mathematics WebsiteMake a Mathematical Model
Represent and solve equations and inequalities graphicallyMCC9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.)MCC9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
EQ: How can graphs of linear or exponential equations be used to solve problems?
Learning Targets:I can …
identify solutions and non-solutions of linear and exponential equations. (REI.10) graph points that satisfy linear and exponential equations. (REI.10) explain why a continuous curve or line contains an infinite number of points on the curve, each representing a solution to the
equation modeled by the curve. (REI.10) approximate or find solutions of a system of two functions (linear and/or exponential) using graphing technology or a table of
values. (REI.11) explain what it means when two curves {y = f(x) and y = g(x)} intersect i.e. what is the meaning of x and what is the meaning of
f(x) = g(x). (REI.11) graph a system of linear equations, find or estimate the solution point, and explain the meaning of the solution in terms of the
system. (REI.11)♦
Concept Overview: Beginning with simple, real-world examples help students to recognize a graph as a set of solutions to an equation. For example, if the equation y = 6x + 5 represents the amount of money paid to a babysitter (i.e., $5 for gas to drive to the job and $6/hour to do the work), then every point on the line represents an amount of money paid, given the amount of time worked.
Explore visual ways to solve an equation such as 2x + 3 = x – 7 by graphing the functions y = 2x + 3 and y = x – 7. Students should recognize that the intersection point of the lines is at (–10, –17). They should be able to verbalize that the intersection point means that when x = -10 is substituted into both sides of the equation, each side simplifies to a value of –17. Therefore, –10 is the solution to the equation. This same approach can be used whether the functions in the original equation are linear, nonlinear or both.
Using technology, have students graph a function and use the trace function to move the cursor along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of the calculator, emphasizing that every point on the curve represents a solution to the equation. Begin with simple linear equations and how to solve them using the graphs and tables on a graphing calculator. Then, advance students to nonlinear situations so they can see that even complex equations that might involve quadratics, absolute value, or rational functions can be solved fairly easily using this same strategy. While a standard graphing calculator does not simply solve an equation for the user, it can be used as a tool to approximate solutions.
Use the table function on a graphing calculator to solve equations. For example, to solve the equation x2 = x + 12, students can examine the equations y = x2 and y = x + 12 and determine that they intersect when x = 4 and when x = –3 by examining the table to find where the y-values are the same.
Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions.
Vocabulary:x - coordinate – the first number in an ordered pairIntersection – the ordered pair or set of elements common to both equations or inequalitiesSolution – a replacement for the variable in an open sentence that results in a true sentenceLinear function – a function that can be written in the form y = mx + b, where x is the independent variable and m and b are real number. Its graph is a line.Sample Problem(s):
Given a graph of the equation x + 3y = 6, find three solutions that will satisfy the equation. (REI10) REI10 Solution
Given a graph representing the growth of a savings account over time with a given rate of return, determine the value of the account after 3 years, 5 years, 10 years, 12 years and 6 months. (REI10)
The cost of producing a soccer ball is modeled by C = 10x + 1000. The sales price of a soccer ball is $20. Explain why a company has to sell 100 soccer balls before they will make a profit. (REI11)
Use technology to graph and compare a beginning salary of $30 per day increased by $5 each day and a beginning salary of $0.01 per day, which doubles each day. When are the salaries equal? How do you know? (REI11)
Standard Topic Resources Teacher Notes
MCC9-12.A.REI.10
How graphs represent solutions
Textbook Section #’s:(Online Textbook Codes) Prentice Hall M3- 11-2,
Student Misconceptions: Students may believe that the graph of a function is simply a line or curve “connecting the dots,” without recognizing that the graph represents all solutions to the equation. Additionally, students may believe that two-variable inequalities have no application in the real world. Teachers can consider business related problems (e.g., linear programming applications) to engage students in discussions of how the inequalities are derived and how the feasible set includes all the points that satisfy the conditions stated in the inequalities.
Probing questions:
1. What do the points on a line or curve represent?2. How do you determine if a point or ordered pair is a solution to an equation?
Differentiation Strategy:1. Have students create a table of value by hand or using a graphing calculator, use the TABLE feature to find points on the curve.2. Have students make a table of values for the given function before plotting points.
Cooperative Learning Strategy: Sage and Scribe How graphs represent solutions
MCC9-12.A.REI.11
Graphical approaches to
solving equations
Textbook Section #’s: Prentice Hall A1-7-1
Additional resources: Matching with Graphs Solving Graphically Notetaking Guide Solving
Graphically Remediation Solving
Graphically
Student Misconceptions: Students may also believe that graphing linear and other functions is an isolated skill, not realizing that multiple graphs can be drawn to solve equations involving those functions.
Probing questions: 1. What is the difference between an independent system and a dependent system?
Differentiation Strategy:1. Use graphing calculators to graph equations to help students understand that the solution, or intersection, does not have to be integer coordinates
Cooperative Learning Strategy: Roundtable Graphical Approach to Solving Equations
Understand the concept of a function and use function notationMCC9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Draw examples from linear and exponential functions.)MCC9-12.F.IF.2 Use function notations, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Draw examples from linear and exponential functions.)MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.)
Interpret functions that arise in applications in terms of the contextMCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ (Focus on linear and exponential functions.)MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.★ (Focus on linear and exponential functions.)MCC9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.)
Analyze functions using different representationsMCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.★ MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.★ MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)
EQ: How can functions be used to represent relationships between quantities?
Learning Targets:I can …
define functions and explain what they are in my own words. (IF1) use function notation, evaluate functions at any point in the domain, give general statements about how f(x) behaves at
different regions in the domain (as x gets very large or very negative, close to 0 etc.), and interpret statements that use function notation. (IF2)
explain the difference and relationship between domain and range. (IF2) state, and/or find, the domain and range of a function from a function equation, table or graph. (IF2) look at data (from a table, graph, or set of points) and determine if the data is a function and explain my conclusion. (IF3) write a function from a sequence or a sequence from a function. (IF3) explain how an arithmetic or geometric sequence is related to its algebraic function notation. (IF3) write arithmetic and geometric sequences recursively. (IF3) write an explicit formula for arithmetic and geometric sequences. (IF3) connect arithmetic sequences to linear functions and geometric sequences to exponential functions and explain those
connections. (IF3) interpret x and y intercepts, where the function is increasing or decreasing, where it is positive or negative, its end behaviors,
given the graph, table or algebraic representation of a function in terms of the context of the function. (Only linear and exponential). (IF4)♦
find and/or interpret appropriate domains and ranges for linear and exponential functions. (IF5) explain the relationship between the domain of a function and its graph in general and/or to the context of the function.
(IF5) calculate and interpret the average rate of change over a given interval of a function from a function equation, graph or
table, and explain what that means in terms of the context of the function. (IF6) estimate the rate of change of a function from its graph at any point in its domain. (IF6)• accurately graph a linear function by hand by identifying key features of the function such as the x- and y-intercepts and
slope. ♦• graph a linear or exponential function using technology. (IF7a)• sketch the graph of an exponential function accurately identifying x- and y-intercepts and asymptotes. (IF7e)• describe the end behavior of an exponential function (what happens as x goes to positive or negative infinity). (IF7e)
• discuss and compare two different functions (linear and/or exponential) represented in different ways (tables, graphs or equations). Discussion and comparisons should include: identifying differences in rates of change, intercepts, and/or where each function is greater or less than the other. (IF9)
Concept Overview: There are five important ideas to consider when thinking about functions.1. Definition – A function is a rule that assigns each element of set A to a unique element of set B.
Thus, a function is a mapping of some element from a domain (set A) into a range (set B). While most of the time we think of the domain and range as being sets of numerical values, this is not always the case. It is important that students understand that a function can operate on non-numerical values.o Example: An amusement park has a sign that displayed in front of the bumper car that says a person must be at
least 4 feet tall to get on the ride. If John is 3 feet 11 inches, the rule assigns him to the group of non-riders. Susan is 4 feet 2 inches, so according to the rule (function) Susan is assigned to the group of riders.
o Example: Applying rigid motion to a triangle. The triangle is the input or domain. The rotations, reflections, translations the triangle is put through are the function “rules” and the final transformed triangle is the output or range. (See Unit 5 & 6 of this course)
2. Covariance and rate of change : The independent and dependent variables of a function have a covariant relationship. Patterns in how the two variables change together, let us know to which family of functions a particular function belongs. Examining the rate of change of a function gives us some important information. (Difference tables are valuable tools for examining rates of change.)
3. Families of Functions : Functions that share the same type of rate of change belong to the same family of functions. A linear function will have a constant rate of change. An exponential function has a rate of change that is proportional to the function value. In this course, students will need to be able to discern between linear and exponential functions. In later courses, students will explore quadratic, exponential and trigonometric functions. Note that sequences – both arithmetic and geometric – can be considered to be functions where the domain is restricted to only the positive integers.
4. Combining and Transforming Functions : Under certain conditions, it is possible to add, subtract, multiply or divide functions, as well as to compose functions together. It is also possible to create transformations of functions in predictable ways. There are patterns in transformations of functions which are consistent across all different families of functions. These are helpful when graphing functions. Under appropriate conditions, functions have inverses which “undo” them.
5. Multiple Representations of Functions : Functions have multiple representations – Algebraic equations, Table, Graph, Verbal descriptions and Context. Students should be comfortable with all representations and be fluent in moving between representations. They should understand that changing the representation does NOT change the function. Each different representation can help students understand a different facet of a function. Certain representations are more useful in certain contexts. Understanding the links between different representations is critical in gaining a deeper understanding of a function. For example: What does the rate of change (slope) of a linear function look like in a table? On a graph? In an algebraic equation? In a verbal description?
It is highly recommended that teachers use technology when teaching this unit – use graphing calculators, Excel or an online graphing utility.
Provide applied contexts in which to explore functions. For example, examine the amount of money earned when given the number of hours worked on a job, and contrast this with a situation in which a single fee is paid by the “carload” of people, regardless of whether 1, 2, or more people are in the car.
Use diagrams to help students visualize the idea of a function machine. Students can examine several pairs of input and output values and try to determine a simple rule for the function.
Rewrite sequences of numbers in tabular form, where the input represents the term number (the position or index) in the sequence, and the output represents the number in the sequence.
Help students to understand that the word “domain” implies the set of all possible input values and that the integers are a set of numbers made up of {…-2, -1, 0, 1, 2, …}.
Distinguish between relationships that are not functions and those that are functions (e.g., present a table in which one of the input values results in multiple outputs to contrast with a functional relationship). Examine graphs of functions and non-functions,
recognizing that if a vertical line passes through at least two points in the graph, then y (or the quantity on the vertical axis) is not a function of x (or the quantity on the horizontal axis).
Flexibly move from examining a graph and describing its characteristics (e.g., intercepts, relative maximums, etc.) to using a set of given characteristics to sketch the graph of a function.
Examine a table of related quantities and identify features in the table, such as intervals on which the function increases, decreases, or exhibits periodic behavior. Recognize appropriate domains of functions in real-world settings. For example, when determining a weekly salary based on hours worked, the hours (input) could be a rational number, such as 25.5. However, if a function relates the number of cans of soda sold in a machine to the money generated, the domain must consist of whole numbers.
Given a table of values, such as the height of a plant over time, students can estimate the rate of plant growth. Also, if the relationship between time and height is expressed as a linear equation, students should explain the meaning of the slope of the line. Finally, if the relationship is illustrated as a linear or non-linear graph, the student should select points on the graph and use them to estimate the growth rate over a given interval.
Explore both linear and exponential function and help students to make connections in terms of general features. Examine multiple real-world examples of exponential functions so that students recognize that a base between 0 and 1 (such as an equation describing depreciation of an automobile [ represents the value of a $15,000 automobile that depreciates 20% per year over the course of x years]) results in an exponential decay, while a base greater than 1 (such as the value of an investment over time [ represents the value of an investment of $5,000 when increasing in value by 7% per year for x years]) illustrates growth. f(x)=15,000(0.8)x
f(x)=5,000(1.07)x
Click here for a more detailed conceptual foundation for this topic. Please note that systems of inequalities is not addressed in this unit.
Vocabulary:Polynomial function – a function of the form f(x) = anxn + an-1xn-1 + …. + a1x + a0 where an ≠0, a0, a1, a2, . . . an are real numbers, and the exponents are all whole numbers.Rational function – a function in the form f(x) = p(x)/g(x) where p(x) and q(x) are polynomials and q(x) ≠ 0. Absolute Value function – a function that contains an absolute value expressionExponential function – a function that involves the expression bx where the base b is a positive number other than 1.
Sample Problem(s): Write a story that would generate a relation that is a function. Write a story that would generate a relation that is not a function. (IF1)
Find a function from science, economics, or sports, write it in function notation and explain its meaning at several points in the domain. (IF2)
Draw the next arrangement of blocks in the sequence and describe the sequence using symbols. (IF3)
Create a story that would generate a linear or exponential function and describe the meaning of key features of the graph as they relate to the story. (IF4)
You are hoping to make a profit on the school play and have determined the function describing the profit to be f(t) = 8t – 2654 where t is the number of tickets sold. What is a reasonable domain for this function? Explain. (IF5)
Create a function in context where the domain would be: (IF5)a) All real numbers.
b) Integers. c) Negative integers. d) Rational Numbers. e) (10, 40).
The graph models the speed of a car. Tell a story using the graph to describe what is happening in various intervals. (IF6)
Graph the linear function that has a slope of 2 and crosses the y-axis at – 3. Write the equation for the function and identify the intercepts. (IF7a) IF7a Solution
The population of salmon in a lake triples each year. The current population is 472. Model the situation graphically. Include the last three years and the next two. Model the situation with a function. (IF7e)
Which has a greater slope? (IF9) f(x) = 3x + 5 A function representing the number of bottle caps in a shoebox where 5 are added each time.
CommentarySlope measures the rate of change in the dependent variable as the independent variable changes. The greater the slope the steeper the line. When the slope, m, is positive, the line slants upward to the right. The more positive m is, the steeper the line will slant upward to the right
SolutionThis linear function: f(x) = mx + b; the slope of the line is m.
o f(x) = 3x + 5, the slope is 3o f(x) = x + 5, is the function that represents the bottle caps in the shoebox, the slope is 1
The second function has a greater slope.
Create a graphic organizer to highlight your understanding of functions and their properties by comparing two functions using at least two different representations. (IF9)
Standard Topic Resources Teacher Notes
MCC9-12.F.IF.1MCC9-12.F.IF.2MCC9-12.F.IF.3
Functions and Function Notation
Textbook Section #’s:(Online Textbook Codes) Prentice Hall A1- 5-2,5-
3,5-4, 8-7,8-8 Prentice Hall M3- 11-1,
11-3 McDougal Littell M1- 3-13
Additional resources: Function Notation Intro (K)
Model Lesson: Function Notation (K)Students will learn about the various representations of relations and functions, including function notation
Student Misconceptions: Students may believe that all relationships having an input and an output are functions, and therefore, misuse the function terminology. Students may also believe that the notation f(x) means to multiply some value f times another value x. The notation alone can be confusing and needs
Functions: Graphs, Tables, Equations (S) Functions with Fiona
careful development. For example, f(2) means the output value of the function f when the input value is 2.
Probing questions: 1. What is the difference between a relation and a function?2. All functions are relations, are all relations functions? Why?3. What does a mapping represent?4. What is the difference between finding the values of linear functions and non-linear functions?5. Sequences are functions with what type of numbers as the domain?6. How can a pattern in a sequence of numbers be represented?7. What can be asked to describe the relationship between each term of a sequence to the ones before it?8. Do you connect the points when graphing a sequence?
Differentiation Strategy:1. Have students say ‘f or x’ when reading the symbol f(x) to help develop the understanding that the solution is a function of x, the input.2. Have students look up the definition of recur, and use the definition to understand the meaning of recursive.3. Use a mapping diagram to allow students to draw arrows to show how each corresponding element from the domain, is paired with each corresponding element from the range.4. After graphing, encourage students to use the vertical line test to determine if a relation is a function.
Additional resources: Functioning Well Comparing Linear and
Exponential Functions Notes on Linear and
Exponential Functions
Student Misconceptions: Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domains. Students may also believe that the slope of a linear function is merely a number used to sketch the graph of the line. In reality, slopes have real-world meaning, and the idea of a rate of change is fundamental to understanding major concepts from geometry to calculus.
Probing questions: 1. How can you find the domain and range of a function from a graph?2. What is always true about the domain and range of exponential functions?3. What do the x-intercept and y-intercept represent in a given scenario?4. What is the relationship between y and f(x) in a function rule?5. How do you write slope in terms of f(x)?6. How can the relationship between the domain and range be represented?
Differentiation Strategy:1. Have students use graphs of ordered pairs to visually identify the type of function formed and then find the equation using the pattern.
2. Allow students to use a graphing calculator to visually identify key features of graphs
Cooperative Learning Strategy: Who Am I? Functions in Context
MCC9-12.F.IF.7aMCC9-12.F.IF.7eMCC9-12.F.IF.9
Graphing and comparing linear and exponential
functions
Textbook Section #’s: Prentice Hall M3- 11-2,
11-5 Prentice Hall A1- 5-1,
8-7,8-8 McDougal-Littell M1-
1.2,1.3 McDougal-Littell M2
4.4,4.5
Additional resources: Linear & Expon’l Summary
and Practice WS (S)(HW from model lesson)
Patterns and Functions: Linear and Exponential (U)(An activity for students to explore fractals and other patterns)
Graphing Linear and Exponential Functions Learning Task
Comparing Linear and Exponential Functions
Notes on Linear and Exponential Functions
Model Lesson: Linear & Exponential Functions Matching (M)Students will find sets of cards that represent the same relationship in a variety of ways (equation, graph, table, verbal description)
Student Misconceptions: Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all functions and their graphs. Students may also believe that skills such as factoring a trinomial or completing the square are isolated within a unit on polynomials, and that they will come to understand the usefulness of these skills in the context of examining characteristics of functions. Additionally, student may believe that the process of rewriting equations into various forms is simply an algebra symbol manipulation exercise, rather than serving a purpose of allowing different features of the function to be exhibited.
Probing questions: What kind of questions can teachers ask that require higher order thinking? What kind of questions can help students when they get stuck? (U)
Differentiation Strategy:List strategies for addressing varying student needs, including readiness levels, learning styles, interests, and/or personal goals. (S)
Cooperative Learning Strategy: Mix ‘n’ Match Graphing and comparing linear and exponential functions
Coordinate Algebra: Unit 3b – Linear and Exponential Functions(3 weeks)
Content Standards: Build a function that models a relationship between two quantitiesMCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★ (Limit to linear and exponential functions.)MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.)MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Limit to linear and exponential functions.)MCC9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★
Build new functions from existing functionsMCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.)
Construct and compare linear, quadratic , and exponential models and solve problems MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★ MCC9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.★ MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.★ MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.★ MCC9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).★ MCC9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★
Interpret expressions for functions in terms of the situation they modelMCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★ (Limit exponential functions to those of the form f(x) = bx + k.)
Standards for Mathematical Practice: 4 Model with mathematics.8 Look for and express regularity in repeated reasoning.
Standards for Mathematical Practice (4, 8)
EQ: How do mathematically proficient students use mathematical models to solve problems? (MP4) How can recognizing repetition or regularity help solve problems more efficiently? (MP8)
Learning Targets:I can …
apply the mathematics I know to solve problems arising in everyday life, society, and the workplace. (MP4) write an equation to describe a situation. (MP4) apply what I know to make assumptions and approximations to simplify a complicated situation, realizing that these may
need revision later. (MP4) identify important quantities in a practical situation. (MP4) map quantity relationships using such tools as diagrams, tables, graphs, and formulas. (MP4) analyze relationships mathematically to draw conclusions. (MP4) interpret my mathematical results in the context of the situation. (MP4) reflect on whether my results make sense, possibly improving the model if it has not served its purpose. (MP4) notice if calculations are repeated, and look both for general methods and for shortcuts. (MP8) maintain oversight of the problem solving process, while also attending to the details. (MP8) continually evaluate the reasonableness of my intermediate results. (MP8)
Concept Overview: MP4 Model with mathematics.Linear and exponential functions often serve as effective models for real life contexts. Teachers who are developing students’ capacity to "model with mathematics" move explicitly between real-world scenarios and mathematical representations of those scenarios. Teachers might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted. One strategy for developing this skill is to pose scenarios with no question, and ask student to complete the statements, “I notice …, I wonder…” For sample scenarios, click here.
MP8 Look for and express regularity in repeated reasoning.In this unit, students will have the opportunity to explore linear and exponential functions using tables. In the Make a Table strategy (which should really be called Make a Table and Look for Patterns) students have the opportunity to explore and talk through patterns they see in repeated calculations. Students are encouraged to look for and describe patterns both horizontally and vertically, as well as to describe what’s happening “over and over again.” Even the simple activity provided in an extension of the Guess and Check activities (courtesy of Dr. Math), in which students do calculations without ever simplifying any steps, help them see and exploit repeated reasoning.
Resources:MP4 Inside Mathematics WebsiteMake a Mathematical Model
Build a function that models a relationship between two quantitiesMCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.★ (Limit to linear and exponential functions.)MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.)MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. (Limit to linear and exponential functions.)MCC9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★
Build new functions from existing functionsMCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.)
EQ: In what ways can functions be combined to create new functions?
Learning Targets:I can …
write a function that describes a linear or exponential relationship between two quantities.(BF1a) combine different functions using addition, subtraction, multiplication, division and composition of functions to create a new
function. (BF1b) write arithmetic and geometric sequences recursively. (BF2) write an explicit formula for arithmetic and geometric sequences. (BF2) connect arithmetic sequences to linear functions and geometric sequences to exponential functions and explain those
connections. (BF2) identify and explain (in words, pictures or with tables) the effect “k” on a graph of f(x) i.e f(x) + k, kf(x), f(kx), and f(x + k). find the value of “k” given the graphs. (BF3) recognize even and odd functions from their graphs and algebraic expressions. (BF3)
Concept Overview: Provide a real-world example (e.g., a table showing how far a car has driven after a given number of minutes, traveling at a uniform speed), and examine the table by looking “down” the table to describe a recursive relationship, as well as “across” the table to determine an explicit formula to find the distance traveled if the number of minutes is known.
Write out terms in a table in an expanded form to help students see what is happening. For example, if the y-values are 2, 4, 8, 16,
they could be written as 2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used multiple times as a factor.
Focus on one representation and its related language – recursive or explicit – at a time so that students are not confusing the formats.
Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function for the cost of each (given the number of miles driven) is known.
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual models to generate sequences of numbers that can be explored and described with both recursive and explicit formulas. Emphasize that there are times when one form to describe the function is preferred over the other.
Use graphing calculators or computers to explore the effects of a constant in the graph of a function. For example, students should be able to distinguish between the graphs of y = 2x, y = 2 2∙ x, y = (½) 2∙ x, y = 2x + 2, and y = 2(x + 2). This can be ½ accomplished by allowing students to work with a single parent function and examine numerous parameter changes to make generalizations.
Distinguish between even and odd functions by providing several examples and helping students to recognize that a function is even if f(–x) = f(x) and is odd if f(–x) = –f(x). Visual approaches to identifying the graphs of even and odd functions can be used as well.
Click here for a more detailed conceptual foundation on this topic.
Vocabulary:Logarithmic function – the inverse of the exponential function y = bx, denoted by y = logb xSystem of Equations – a set of two equations that can be written in the form of Ax + By = C and Dx + Ey = F where x and y are variables, A and B are not both zero, and D and E are not both zero. Substitution property – If a = b, then a may be replaced by b.
Sample Problem(s): Anne is shopping and finds a $30 sweater on sale for 20% off. When she buys the sweater, she must also pay 6% sales tax. Write an expression for the final price of the sweater in such a way that the original price is still evident. (Extension: If the clerk just adds 14% will the price be correct?) (BF1)
Find an expression, process or calculation to determine the number of squares needed to make the next three patterns in the series. (BF1)
You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Express the amount remaining to be paid off as a function of the number of months, using a recursion equation. (BF1)
A cup of coffee is initially at a temperature of 93º F. The difference between its temperature and the room temperature of 68º F decreases by 9% each minute. Write a function describing the temperature of the coffee as a function of time. (BF1)
The radius of a circular oil slick after t hours is given in feet by , for 0 ≤ t ≤ 10. Find the area of the oil slick as a function of time. (BF1)
Write two formulas that model the pattern: 3, 9, 27, 81…… (BF2)
Commentary
Functions can be defined explicitly, by a formula in terms of the variable. We can also define functions recursively, in terms of the same function of a smaller variable. In this way, a recursive function "builds" on itself. A recursive formula may list the first two (or more) terms as starting values, depending upon the nature of the sequence. In such cases, the an portion of the formula is dependent upon the previous two (or more) terms.
Explicit formula: Recursive formula: Certain sequences, such as this geometric sequence, can be represented in more than one manner. This sequence can be represented as either an explicit formula or a recursive formula.
Continue the pattern for two more iterations graphically and then find a recursive or explicit formula to model the situation. (BF2)
There are 2,500 fish in a pond. Each year the population decreases by 25 percent, but 1,000 fish are added to the pond at the end of the year. Find the population in five years. Also, find the long-term population. (BF2)
Graph the following on a single set of axes: (BF3) BF3 Solution
Compare and contrast the characteristics of these graphs.
Describe the effect of varying the parameters a, h, and k on the shape and position of the graph f(x) = ab(x + h) + k., orally or in written format. What effect do values between 0 and 1 have? What effect do negative values have? (BF3)
Standard Topic Resources Teacher Notes
MCC9-12.F.BF.1
MCC9-12.F.BF.1a
MCC9-12.F.BF.1b
MCC9-12.F.BF.2
Writing equations for
linear and exponential
functions
Textbook Section #’s:(Online Textbook Codes) Prentice Hall M3- 11-1,
11-6 Prentice Hall A1- 5-4, 5-5,
5-7, 8-6, 8-7, 8-8 McDougal-Littell M2-
4.2,4.4,4.5
Additional resources:
Student Misconceptions: Students may believe that the best (or only) way to generalize a table of data is by using a recursive formula. Students naturally tend to look “down” a table to find the pattern but need to realize that finding the 100th term requires knowing the 99th term unless an explicit formula is developed. Students may also believe that arithmetic and geometric sequences are the same. Students need experiences with both types of sequences to be able to recognize the difference and more readily develop formulas to describe them. Additionally, advanced students who
Arithmetic Sequence Recursive Sequence Writing Linear Functions Writing Exponential
Functions Application Writing Exponential
Functions Arithmetic Sequence WS
study composition of functions may misunderstand function notation to represent multiplication (e.g., f(g(x)) means to multiply the f and g function values).
Probing questions: 1. What do you know about the difference between consecutive terms in an arithmetic sequence?2. How can you use the first term in a sequence and the common difference to write a rule, or determine an explicit expression?3. Explain the differences between a recursive rule of a function and an explicit rule of a function.
Differentiation Strategy:1. Have students use graphs of ordered pairs to visually identify the type of function formed and then find the equation using the pattern.2. Use graphing calculators or a spreadsheet application to help students see how the terms will change without performing calculations.3. Use the general form of an explicit and recursive formula and let students substitute needed parameters form the given scenario.
Cooperative Learning Strategy: Sage and Scribe Writing equations of linear and exponential functions
Literacy Strategy: Vocabulary Cut and Paste. (K)
MCC9-12.F.BF.3
Effect of transformations
on functions
Textbook Section #’s: McDougal-Littell M2-
4.4,4.5
Additional resources: High Functioning
(from state frameworks) Transformations of
Exponential Functions Transformation of Linear
Functions
Student Misconceptions: Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by hand and on a graphing calculator to overcome this misconception. Students may also believe that even and odd functions refer to the exponent of the variable, rather than the sketch of the graph and the behavior of the function.
Probing questions: 1. How are the horizontal asymptotes and the y-intercepts affected during transformations>2. How does using the ZOOM feature on a graphing calculator to enlarge a part of the graph effect your answer?
Differentiation Strategy:1. Have students make a table of values for each function along with the parent function. Have students find an ordered pair for the parent function that have the same y-value when first determining horizontal shift directions
Cooperative Learning Strategy: Timed Pair Share Effect of transformations on functions
Construct and compare linear, quadratic , and exponential models and solve problems MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★ MCC9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.★ MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.★ MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.★ MCC9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).★ MCC9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★
Interpret expressions for functions in terms of the situation they modelMCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★ (Limit exponential functions to those of the form f(x) = bx + k.) EQ: How can linear and exponential functions be used to represent problems in real life contexts?
Learning Targets:I can …
• distinguish between situations that can be modeled with linear and exponential functions. (LE1)• prove (using words, pictures, numbers and/or difference tables) that linear functions grow by equal differences over equal
intervals. (LE1a)• prove (using words, pictures, numbers and/or difference table) that exponential functions grow by equal factors over equal
intervals. (LE1a)• recognize situations with a constant rate of change. (LE1b)• recognize situations that can be modeled linearly or exponentially and describe the rate of change per unit as constant or the
growth factor as a constant percent. (LE1c)• recognize situations in which a quantity either grows or decays by a constant percent rate. (LE1c)• construct a linear function given an arithmetic sequence, a graph, a description of a relationship or a table of input-output
pairs. (LE2)• construct an exponential function given a geometric sequence, a graph, a description of a relationship or a table of input-
output pairs. (LE2)• explain why a quantity increasing exponentially will eventually exceed a quantity increasing linearly. (LE3)• interpret and explain the parameters in an exponential function in terms of a given context (authentic situation,• graph, symbolic representation.) (LE5)
Concept Overview: Compare tabular representations of a variety of functions to show that linear functions have a first common difference (i.e., equal differences over equal intervals), while exponential functions do not (instead function values grow by equal factors over equal x-intervals). Apply linear and exponential functions to real-world situations. For example, a person earning $10 per hour experiences a constant rate of change in salary given the number of hours worked, while the number of bacteria on a dish that doubles every hour will have equal factors over equal intervals. Provide examples of arithmetic and geometric sequences in graphic, verbal, or tabular forms, and have students generate formulas and equations that describe the patterns. Use a graphing calculator or computer program to compare tabular and graphic representations of exponential and polynomial functions to show how the y (output) values of the exponential function eventually exceed those of polynomial functions. Have students draw the graphs of exponential and other polynomial functions on a graphing calculator or computer utility and examine the fact that the exponential curve will eventually get higher than the polynomial function’s graph. A simple example would be to compare the graphs (and tables) of the functions y = x2 and y = 2x and to find that the y values are greater for the exponential function when x > 4.
Students can investigate functions and graphs modeling different situations involving simple and compound interest. Students can compare interest rates with different periods of compounding (monthly, daily) and compare them with the corresponding annual percentage rate. Spreadsheets and applets can be used to explore and model different interest rates and loan terms.
Students can use graphing calculators or programs, spreadsheets, or computer algebra systems to construct linear and exponential functions.
Use real-world contexts to help students understand how the parameters of linear and exponential functions depend on the context. For example, a plumber who charges $50 for a house call and $85 per hour would be expressed as the function y = 85x + 50, and if the rate were raised to $90 per hour, the function would become y = 90x + 50. On the other hand, an equation of y = 8,000(1.04)x could model the rising population of a city with 8,000 residents when the annual growth rate is 4%. Students can examine what would happen to the population over 25 years if the rate were 6% instead of 4% or the effect on the equation and function of the city’s population were 12,000 instead of 8,000.
Graphs and tables can be used to examine the behaviors of functions as parameters are changed, including the comparison of two functions such as what would happen to a population if it grew by 500 people per year, versus rising an average of 8% per year over the course of 10 years.
Sample Problem(s): An accountant has two ways of depreciating equipment. One way is to depreciate by a fixed amount each year. The other way is to depreciate by a fixed percentage each year. Which depreciation method is linear? Which depreciation method is exponential? (LE1)
A cell phone company has three plans. Graph the equation for each plan, and analyze the change as the number of minutes used increases. When is it beneficial to enroll in Plan 1? Plan 2? Plan 3? (LE1)
1. $59.95/month for 700 minutes and $0.25 for each additional minute,2. $39.95/month for 400 minutes and $0.15 for each additional minute, and3. $89.95/month for 1,400 minutes and $0.05 for each additional minute.
A computer store sells about 200 computers at the price of $1,000 per computer. For each $50 increase in price, about ten fewer computers are sold. How much should the computer store charge per computer in order to maximize their profit? (LE1)
A couple wants to buy a house in five years. They need to save a down payment of $8,000. They deposit $1,000 in a bank account earning 3.25% interest, compounded quarterly. How much will they need to save each month in order to meet their goal? (LE1)
Sketch and analyze the graphs of the following two situations. What information can you conclude about the types of growth each type of interest has? (LE1)
o Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a year, but she does not compound the interest.
o Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest compounded annually.
Create a story that demonstrates one quantity changing at a constant rate per unit interval relative to another. (LE1)
Write a function that models the population of Smithville, a town that in 2003 was estimated to have 35,000 people that increases by 2.4% every year. Describe a reasonable way to use your function to predict future population in Smithville. (LE2)
Determine an exponential function of the form f(x) = abx using data points from the table. Graph the function and identify the key characteristics of the graph. (LE2) LE2 Solution
x f(x)0 11 33 27
Sara’s starting salary is $32,500. Each year she receives a $700 raise. Write a sequence in explicit form to describe the situation. (LE2)
CommentaryFunctions can be defined explicitly, by a formula in terms of the variable.
SolutionThe explicit formula that represents the situation is , where her annual salary is a function of the number of years she works. Substituting the number of years x, starting with 1, the sequence is 33250, 34000, 34750, 35500, 36250, 37000…
What’s the better deal, earning $1000 a day for the rest of your life or earning $.01 the first day, and doubling it every day for the rest of your life? How do you know? Do you think an 80-year-old would make the same choice? Should she? (LE3)
Contrast the growth of the f(x)=x3 and f(x)=3x. (LE3)
Annie is picking apples with her sister. The number of apples in her basket is described by n = 22t + 12, where t is the number of minutes Annie spends picking apples. What do the numbers 22 and 12 tell you about Annie’s apple picking? (LE5)
CommentaryA practical application of slope, m, is a rate. A rate describes how much one variable changes with respect to another. Rates are often used to describe relationships between time and an action. This equation is called the slope-intercept form for a line. The changes occur as a function of time. The slope is m and the y-intercept is b. The point where the graph crosses the y-axis is called the y-intercept. The y-intercept represents an initial condition, or what action is occurring when the time is zero.
Solution This linear function n = 22t + 12, is written in the general form y = mx + b. The slope of the line is m. In the formula, 22 is the slope and represents the number of apples Annie picks in 1 minute. In the formula, 12 represents the initial amount of apples in the basket. As she adds 22 apples to the basket each minute, that amount is automatically increased by the 12 apples in the basket before she began.
A function of the form f(n) = P(1 + r)n is used to model the amount of money in a savings account that earns 5% interest, compounded annually, where n is the number of years since the initial deposit. What is the value of r? What is the meaning of the constant P in terms of the savings account? Explain either orally or in written format. (LE5)
Standard Topic Resources Teacher Notes
MCC9-12.F.LE.1
MCC9-12.F.LE.1a
MCC9-12.F.LE.1b
MCC9-12.F.LE.1c
MCC9-12.F.LE.2
MCC9-12.F.LE.3
Comparing linear and exponential
models
Additional resources: Building and Combining
Functions(from state frameworks)
Community Service Sequences and Functions(from state frameworks)
PPT Exponential Models Exponential Worksheet
Student Misconceptions: Students may believe that all functions have a first common difference and need to explore to realize that, for example, a quadratic function will have equal second common differences in a table. Students may also believe that the end behavior of all functions depends on the situation and not the fact that exponential function values will eventually get larger than those of any other polynomial functions.
Probing questions: What kind of questions can teachers ask that require higher order thinking? What kind of questions can help students when they get stuck? (U)
Differentiation Strategy:List strategies for addressing varying student needs, including readiness levels, learning styles, interests, and/or personal goals. (S)
Cooperative Learning Strategy: Flash Card Game Comparing Linear and exponential models
MCC9- MCC9-12.F.LE.5
Interpreting function
expressions in context
Additional resources: You’re Toast Dude
(from state frameworks) Linear In Context PPT Exponential Models
Student Misconceptions: Students may believe that changing the slope of a linear function from “2” to “3” makes the graph steeper without realizing that there is a real-world context and reason for examining the slopes of lines. Similarly, an exponential function can appear to be abstract until applying it to a real-world situation involving population, cost, investments, etc.
Probing questions: What kind of questions can teachers ask that require higher order thinking? What kind of questions can help students when they get stuck? (U)
1. Have students enter original function and translated functions in graphing calculator to observe shifts, stretches, and shrinks.2. Have students use tracing paper to trace the graph of original function and use the traced graph to model shifts.
Cooperative Learning Strategy: Timed Pair Share Interpreting function expressions in context
Literacy Strategy: Vocabulary Who Has I Have . (K)Unit # Summative Assessment 3B (3A is above)
