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) =bx + k, where b1, 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; 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 representations
MCC9-12.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 quantities
MCC9-12.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 functions
MCC9-12.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 model
MCC9-12.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 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; 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 representations
MCC9-12.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)
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 Tablestrategy (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 Website
Make a Mathematical Model
Diagnostic: Prerequisite Assessment 3a
Solving Equations Graphically
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.★
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)
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 pair
Intersection – the ordered pair or set of elements common to both equations or inequalities
Solution – a replacement for the variable in an open sentence that results in a true sentence
Linear 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 equationx + 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)