Unit 3
Functions

 

Algebra II

Unit Topic and Length:

45 days

 

In this unit, students extend their knowledge of linear, quadratic, polynomial, rational, radical and trigonometric functions learned in Algebra I and the first half of Algebra II, and use them in modeling situations.  The focus for this unit is on exponential and logarithmic functions.

 

In Unit 3 students synthesize and generalize what they have learned about a variety of function families.  They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function.  They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgements about the domain over which a model is a good fit.  (Mathematics Appendix A, p.36-42, with adjustments)

 

·         Extend the properties of exponents to rational exponents.

·         Reason quantitatively and use units to solve problems.

·         Write expressions in equivalent forms to solve problems.

·         Create equations that describe numbers or relationships.

·         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 the 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.

 

Standards:

 

N-RN.A.1  Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.  For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

 

N-RN.A.2  Rewrite expressions involving radicals and rational exponents using the properties of exponents.

 

  • Standards Clarification:  Including expressions where either base or exponent may contain variables.

 

N-Q.A.2  Define appropriate quantities for the purpose of descriptive modeling.

 

  • Standards Clarification:  This standard will be assessed in Algebra II by ensuring that some modeling tasks (involving Algebra II content or securely held content from previous grades and courses) require the student to create a quantity of interest in the situation being described (i.e., this is not provided in the task). For example, in a situation involving periodic phenomena, the student might autonomously decide that amplitude is a key variable in a situation, and then choose to work with peak amplitude.  .

 

A-SSE.B.3c  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%.

 

  • Standards Clarification:  Tasks have a real-world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation. In Algebra II, tasks include exponential expressions with rational or real exponents.

 

A-SSE.B.4  Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

 

  • Standards Clarification:  This standard includes using the summation notation symbol.

 

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.

 

  • Standards Clarification:  Tasks have a real-world context. In Algebra II, tasks include exponential equations with rational or real exponents, rational functions, and absolute value functions, but are constrained to simple cases.

 

A-REI.D.11  Explain why the xcoordinates 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.

 

  • Standards Clarification:  In Algebra II, tasks may involve any of the function types mentioned in the standard.

 

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 f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ≥ 1.

 

  • Standards Clarification:  This standard is Supporting Content in Algebra II. This standard should support the Major Content in F-BF.2 for coherence.

 

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.

 

  • Standards Clarification:  Tasks have a real-world context. In Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions.  Emphasize the selection of a model function based on behavior of data and context.

 

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.

 

  • Standards Clarification:  Tasks have a real-world context. In Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions.  Emphasize the selection of a model function based on behavior of data and context.

 

F-IF.C.7e  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

e.  Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

 

  • Standards Clarification:  Focus on application and using key features to guide the selection of the appropriate type of model function.

 

F-IF.C.8b  Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

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

 

  • Standards Clarification:  Tasks include knowing and applying A = Pert and A = P (1 + r/n)nt .

 

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.

 

  • Standards Clarification:  In Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions.

 

F-BF.A.1a  Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

 

  • Standards Clarification:  Tasks have a real-world context. In Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. 

 

F-BF.A.1b  Write a function that describes a relationship between two quantities.

b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

 

  • Standards Clarification:  Combining functions also includes composition of functions.  Include all types of functions studied.  Develop models for more complex or sophisticated situations than in previous courses.

 

F-BF.A.2  Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

 

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.

 

  • Standards Clarification:  In Algebra II, tasks may involve simple radical, rational, polynomial, exponential, logarithmic, and trigonometric functions. Tasks may involve recognizing even and odd functions.  Note the effect of multiple transformations on a single graph and emphasize the common effect of each transformation across function types.

 

F-BF-B.4a  Find inverse functions.

a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x1) for x ≠ 1.

 

  • Standards Clarification:  Extend to simple rational, simple radical, and simple 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 inputoutput pairs (include reading these from a table).

 

  • Standards Clarification:  In Algebra II, tasks will include solving multi-step problems by constructing linear and exponential functions.

 

F-LE.A.4  For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

 

  • Standards Clarification:  Students learn terminology that logarithm without a base specified is base 10 and that natural logarithm always refers to base e.  Recognize and use logarithms as solutions for exponentials.

 

F-LE.B.5  Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.

 

  • Standards Clarification:  Tasks have a real-world context. In Algebra II, tasks include exponential functions with domains not in the integers.

 

 

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.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

N-RN.A.1

·         I can define radical notation as a convention used to represent rational exponents.

·         I can explain the properties of operations of rational exponents as an extension of the properties of integer exponents.

·         I can explain how radical notation, rational exponents, and properties of integer exponents relate to one another.

 

N-RN.A.2

·         I can, using the properties of exponents, rewrite a radical expression as an expression with a rational exponent.

·         I can, using the properties of exponents, rewrite an expression with a rational exponent as a radical expression.

 

N-Q.A.2

·         I can define descriptive modeling.

·         I can determine appropriate quantities for the purpose of descriptive modeling.

A-SSE.B.3c

·         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.

 

A-SSE.B.4

·         I can find the first term in a geometric sequence given at least two other terms.

·         I can define a geometric series as a series with a constant ratio between successive terms.

·         I can use the formula S = ((a (1-rn))/ (1-r)) to solve problems.

 

A-CED.A.1

·         I can solve equations specified in this standard in one variable.

·         I can solve inequalities specified in this standard in one variable.

·         I can describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve.

·         I can create equations and inequalities specified in this standard in one variable and use them to solve problems.

·         I can create equations and inequalities specified in this standard in one variable to model real-world situations.

·         I can compare and contrast problems that can be solved by the different types of equations specified in this standard.

 

A-REI.D.11

·         I can approximate or find the solutions to a system involving any of the function types listed in this standard.

·         I can explain why the solution to a system involving any of the function types listed in this standard will occur at the point(s) of intersection.

 

F-IF.A.3

·         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.

F-IF.B.4

·         I can identify the key features from a table or graph of polynomial, exponential, logarithmic, and trigonometric functions.

·         I can sketch functions that model key feature behavior.

 

F-IF.B.6

·         I can calculate the average rate of change of a given interval for polynomial, exponential, logarithmic, and trigonometric functions.

·         I can demonstrate that the average rate of change of a non-linear function is different for different intervals.

 

F-IF.C.7e

·         I can graph polynomial, exponential, logarithmic, and trigonometric functions.

·         I can describe key features of polynomial, exponential, logarithmic, and trigonometric functions.

 

F-IF.C.8b

·         I can identify how key features of an exponential function relate to characteristics in a real-world context.

·         I can write and classify real-world problems as an exponential growth or decay.

·         I can use and applying A = Pert and A = P (1 + r/n)nt .

 

F-IF.C.9

·         I can compare key features of two representations of functions (polynomial, exponential, logarithmic, and trigonometric).

 

F-BF.A.1a

·         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 (involving polynomial, exponential, logarithmic, and trigonometric functions).

 

F-BF.A.1b

·         I can combine two functions (all functions studied) using the operations of addition, subtraction, multiplication, and division.

·         I can evaluate the domain of the combined function (all functions studied).

·         I can build standard functions (all functions studied) to represent relevant relationships/quantities given a real-world situation or mathematical process.

·         I can determine which arithmetic operation should be performed to build the appropriate combined function (all functions studied) given a real-world situation or mathematical process.

·         I can relate the combined function (all functions studied) to the context of the problem.

 

F-BF.A.2

·         I can identify arithmetic and geometric patterns in given sequences.

·         I can generate arithmetic and geometric sequences from recursive and explicit formulas.

·         I can, given an arithmetic or geometric sequence in recursive form, translate into the explicit formula.

·         I can, given an arithmetic or geometric sequence as an explicit formula, translate into the recursive form.

·         I can use given and constructed arithmetic and geometric sequences, expresses both recursively and with explicit formulas, to model real-life situations.

·         I can determine the recursive rule given arithmetic and geometric sequences.

·         I can determine the explicit formula given arithmetic and geometric sequences.

·         I can justify the translation between the recursive form and explicit formula for arithmetic and geometric sequences.

 

F-BF.B.3

·         I can perform the tasks below which may involve simple radical, rational, polynomial, exponential, logarithmic, and trigonometric functions.

·         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 transformations of a function on a graph.

·         I can describe the effects of transformations on parent functions.

 

F-BF-B.4a

·         I can define inverse function (this extends to simple rational, simple radical, and simple exponential functions).

·         I can solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse (this extends to simple rational, simple radical, and simple exponential functions).

 

F-LE.A.2

·         I can recognize arithmetic sequences can be expressed as linear functions.

·         I can recognize geometric sequences can be expressed 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.

 

F-LE.A.4

·         I can use the properties of logs.

·         I can describe the key features of logs.

·         I can use logarithmic form to solve exponential models.

·         I can recognize that logarithm without a base specified is base 10 and that natural logarithm always refers to base e. 

·         I can recognize and use logarithms as solutions for exponentials.

 

F-LE.B.5

·         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.

·         I can interpret the parameters in a linear or exponential function in terms of a context.

·         I can interpret the parameters of exponential functions with domains not in the integers.

 

Enduring Understandings:

 

·         A function is a relationship between variables in which each value of the input variable is associated with a unique value of the output variable.

·         Various mathematical situations require various functions to model the data.

·         The basic modeling cycle can be used to shed light on mathematical structures.

·         Adjusting the parameters of a situation can improve the model.

·         Solving exponential and logarithmic equations and graphing exponential and logarithmic functions can be approached by hand and by using technology.

·         Comparing the speed at which a function increases can help determine which type of function best models the data.

·         They will comprehend the meaning of a logarithm of a number and know when to use logarithms to solve exponential functions.

·         Transformations on a graph always have the same effect regardless of the type of underlying function.

·         Transforming functions is essential to the interpretation of data.

·         It is important to be able to translate easily among the equation of a function, its graph, its verbal representation, and its numerical representation.

 

Essential Questions:

 

·         How are exponents and logarithms related?

·         What is the relationship between exponential functions and logarithms?

·         How do you model a quantity that changes regularly over time by the same percentage?

·         How can the modeling cycle help determine which type of function best models a given set of data?

·         How can data be used to determine the function that fits it best?

·         What can be learned from studying models of exponential growth and decay?

·         How can studying families of functions be beneficial to interpreting graphs and data?

·         What are some of the characteristics of the graph of an exponential function?

·         What are some of the characteristics of the graph of a logarithmic function?

·         How can you use properties of exponents to derive properties of logarithms?

·         How can you write a rule for the nth term of a sequence?

·         How can you recognize an arithmetic sequence from its graph?

·         How can you recognize a geometric sequence from its graph?