Linear & Exponential Functions
Unit Length and Description:
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 explore systems of equations and inequalities, and they find and interpret their solutions. 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 function. (Mathematics Appendix A, p.19)
· Represent and solve equations and inequalities graphically
· Understand the concept of a function and use function notation
· Interpret functions that arise in applications in terms of a context.
· Analyze functions using different representations.
· Build a function that models a relationship between two quantities
· Build new functions from existing functions.
· Construct and compare linear, quadratic, and exponential models and solve problems.
· Interpret expressions for functions in terms of the situation they model.
· Write expressions in equivalent forms to solve problems.
· Create equations that describe numbers or relationships.
A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★
A-REI.D.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 (𝑥) and/or (𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t≈1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
F-BF.A.1 Write a function that describes a relationship between two quantities.★
F-BF.B.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.
F-IF.A.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 (𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦=(𝑥).
F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by (0)= 𝑓(1)=1, 𝑓(𝑛+1)=𝑓(𝑛)+𝑓(𝑛−1) for 𝑛≥1.
F-IF.B.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.★
F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛) gives the number of person-hours it takes to assemble 𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function.★
F-IF.B.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.★
F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★
F-LE.A.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).★
F-LE.A.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.★
F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.★
Focus Standards of Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
· I can solve quadratic equations in one variable
· I can solve quadratic inequalities in one variable
· I can create quadratic equations and inequalities in one variable and use them to solve problems
· I can create quadratic equations and inequalities in one variable to model real-world situations
· I can recognize and use function notation to represent linear and exponential equations
· I can recognize that if (x1 , y1) and (x2 , y2) share the same location in the coordinate plane that x1 = x2 and y1 = y2
· I can recognize that f(x) = g(x) means that there may be particular inputs of f and g for which the outputs of f and g are equal
· I can explain why the x-coordinates of the points where the graph of the equations y=f(x) and y=g(x) intersect are the solutions of the equations f(x) = g(x). (Include cases where f(x) and/or g(x) are linear and exponential equations)
· I can approximate/find the solution(s) using an appropriate method for example, 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 and exponential equations)
· I can use the properties of exponents to transform simple expressions for exponential functions
· I can use the properties of exponents to transform expressions for exponential functions
· I can choose and produce an equivalent form of an exponential expression to reveal and explain properties of the quantity represented by the original expression
· I can explain the properties of the quantity or quantities represented by the transformed exponential expression
· I can define “explicit function” and “recursive process”
· I can write a function that describes a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
· I can identify the effect a single transformation will have on the function (symbolic or graphic)
· I can use technology to identify effects of single transformations on graphs of functions
· I can graph a given function by replacing f(x) with f(x)+k, kf(x), f(kx), or f(x+k) for specific values of k (both positive and negative)
· I can describe the differences and similarities between a parent function and the transformed function
· I can find the value of k, given the graphs of a parent function, f(x), and the transformed function: f(x)+k, kf(x), f(kx), or f(x+k)
· I can recognize even and odd functions from their graphs and from their equations
· I can experiment with cases and illustrate an explanation of the effects on the graph using technology
· I can identify the domain and range of a function
· I can determine if a relation is a function
· I can determine the value of the function with proper notation (i.e. f(x) = y, the y value is the value of the function at a particular value of x)
· I can evaluate functions for given values of x
· I can identify mathematical relationships and express them using function notation
· I can define a reasonable domain, which depends on the context and/or mathematical situation, for a function focusing on linear and exponential functions
· I can evaluate functions at a given input in the domain, focusing on linear and exponential functions
· I can interpret statements that use functions in terms of real world situations, focusing on linear and exponential functions
· I can recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined by f(0)=f(1)=1, f(n+1) = f(n) + f(n-1) for n≥1
· I can define and recognize the key features in tables and graphs of linear and exponential functions: intercepts; intervals where the function is increasing, decreasing, positive, or negative, and end behavior
· I can identify whether the function is linear or exponential, given its table or graph
· I can interpret key features of graphs and tables of function in the terms of the contextual quantities the function represents
· I can sketch graphs showing key features of a function that models a relationship between two quantities from a given verbal description of the relationship
· I can, given the graph or a verbal/written description of a function, identify and describe the domain of the function
· I can identify an appropriate domain based on the unit, quantity , and type of the function it describes
· I can relate the domain of the function to its graph and, where applicable, to the quantitative relationship it describes
· I can explain why a domain is appropriate for a given situation
· I can recognize slope as an average rate of change
· I can calculate the average rate of change of a function (presented symbolically or as a table) over a specified interval
· I can estimate the rate of change from a linear or exponential graph
· I can interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval
· I can graph linear functions by hand in simple cases or using technology for more complicated cases and show/label intercepts of the graph
· I can identify types of functions based on verbal, numerical, algebraic, and graphical descriptions and state key properties (e.g. intercepts, growth rates, average rates of change, and end behaviors)
· I can differentiate between exponential and linear functions using a variety of descriptors (graphically, verbally, numerically, and algebraically)
· I can use a variety of function representations algebraically, graphically, numerically in tables, or by verbal descriptions) to compare and contrast properties of two functions
· I can recognize that linear functions grow by equal differences over equal intervals
· I can recognize that exponential functions grow by equal factors over equal intervals
· I can distinguish between situations that can be modeled with linear functions and with exponential functions to solve mathematical and real-world problems
· I can prove that linear functions grow by equal differences over equal intervals
· I can prove that exponential functions grow by equal factors over equal intervals
· I can recognize situations in which one quantity changes at a constant rate per unit (equal differences) interval relative to another to solve mathematical and real-world problems
· I can recognize situations in which a quantity grows or decays by a constant percent rate per unit (equal factors) interval relative to another to solve mathematical and real-world problems
· I can recognize arithmetic sequences can be expressed as linear functions
· I can recognize geometric sequences can be expresses as exponential functions
· I can construct linear functions, including arithmetic sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)
· I can construct exponential functions, including geometric sequences, given a graph, a description of relationship, or two input-output pairs (include reading these from a table)
· I can determine when a graph, a description of a relationship, or two input-output pairs (include reading these from a table) represents a linear or exponential function in order to solve problems
· I can informally define the concept of “end behavior”
· I can compare tables and graphs of linear and exponential functions to observe that a quantity increasing exponentially exceeds all others to solve mathematical and real-world problems
· I can recognize the parameters in a linear or exponential function including: vertical and horizontal shifts, vertical and horizontal dilations
· I can recognize rate of change and intercept as “parameters” in linear or exponential functions
can interpret the parameters in a linear or exponential function in terms of
· Algebra is the language of symbols, operations, and relationships that allow us to understand many real world phenomena.
· There are many times in real life when one quantity depends on another.
· Real world situations can be represented symbolically, numerically, verbally, and graphically.
· New functions can be created from parent functions through a series of transformations.
· Real world phenomena often involve non-linear relationships.
· Non-constant rates of change denote non-linear relations.
· Mathematical rules that reflect recurring patterns facilitate efficiency in problem solving.
· When analyzing linear and exponential functions, different representations may be used based on the situation presented.
· Algebraic representations can be used to generalize patterns in mathematics.
· Relationships between quantities can be represented symbolically, numerically, graphically, and verbally in the exploration of real world situations.
· The characteristics of radical functions and its representations are useful in solving real-world problems.
· Families of functions exhibit properties and behaviors that can be recognized across representations.
· Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.
· Mathematical functions are relationships that assign each member of one set to a unique member of another set and the relationship is recognizable across representations.
· What are some ways I can represent the relationship between quantities?
· When quantities share a relationship, how can I use that relationship to determine reasonable values for the quantities?
· How do I know when one quantity depends on another
· How can the characteristics of a function be determined from a graph or a given function?
· Why are relations and functions represented in multiple ways?
· How are properties of function and functional operations useful?
· What types of real-life situations involve non-constant rates of change?
· How do linear and non-linear relationships differ? How are they the same?
· How can I use patterns to establish relationships that will help make decisions in real-life situations?
· How can a new function be created from an existing function?
· What characteristics of problems would determine how to model the situation and develop a problem solving strategy?
· How do radical functions model real-world problems and their solutions?
· How are expressions involving radicals and exponents related?
· How can an equation, table, and graph be used to analyze the rate of change and other applicable information, related to a real-world problem and the representative function?
· How can a given function be represented graphically, within a table, by an equation, and in the real-world?
· What connections can be made between various functions and various representations of functions?