ALEX Resources

   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

This lesson provides a review of evaluating functions and finding function rules as well as an introduction to the composition of functions. The review is accomplished through the use of an online exploration using a function machine. The idea of a function machine is also used to explain the composition of functions. Instruction is provided in finding the composition using several different representations of functions (input/output tables, graphs, and function rules). 

This lesson results from the ALEX Resource Gap Project.

   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

Students will be motivated to learn how to build new linear functions from existing linear functions.  Students will bring pictures of themselves and their parents from home to personally involve them in the lesson.  Students will learn to use the patterns inherent in functions to quickly and accurately graph linear functions.  This lesson will only deal with vertical shifts and the steepness of the line.  Horizontal shifts will be dealt with in future lessons. In addtion, in future lessons  students will transfer this knowledge to also graph exponential, quadratic, and absolute value functions.

This is a College- and Career-Ready Standards showcase lesson plan.

   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

In Module 3, Topic A, students both review what they already know about exponential expressions and functions with integer exponents and extend the meaning of an exponential expression to allow for first rational and then irrational exponents (N-RN.1). Students practice applying properties of exponents to expressions with non-integer exponents (N-RN.2). Students use the average rate of change and repeated reasoning to discover Euler’s number, e (F-IF.6). 

   View Standards     Standard(s): [MA2015] AL2 (9-12) 20 :

20 ) 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-CED1]

[MA2015] AL2 (9-12) 36 :

36 ) For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology. [F-LE4]

[MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

At the beginning of Module 3, Topic B, students apply the properties of exponents to solve exponential equations numerically (F-BF.1a) as a way to motivate the need for logarithms, which are first introduced by the more intuitive name “WhatPower." In the intermediate lessons, students discover the logarithmic properties by creating and examining logarithmic tables and answering sets of directed questions. In the final lessons in Topic B, they solve logarithmic equations by applying the inverse relationship between exponents and logarithms (F-LE.4).

   View Standards     Standard(s): [MA2015] AL2 (9-12) 35 :

35 ) Find inverse functions. [F-BF4]

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. [F-BF4a]

Example: f(x) =2x³ or f(x) = (x+1)/(x-1) for x ≠ 1.

[MA2015] AL2 (9-12) 29 :

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

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.

[MA2015] AL2 (9-12) 36 :

36 ) For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology. [F-LE4]

[MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

[MA2019] AL1-19 (9-12) 23 :

23. Identify the effect on the graph of replacing f(x) by f(x)+k,k·f(x), f(k·x), 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 explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.

[MA2019] AL1-19 (9-12) 30 :

30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.

In Module 3, Topic C, students graph logarithmic functions, identifying key features (F-IF.4, F-IF.7) and discover how the logarithmic properties are evidenced in the graphs of corresponding logarithmic functions. The inverse relationship between an exponential function and its corresponding logarithmic function is made explicit (F-BF.3).  In the final lesson in Topic C, students synthesize what they know about linear, quadratic, sinusoidal, and exponential functions to determine which function is most appropriate to use to model a variety of real-world scenarios (F-BF.1a).

   View Standards     Standard(s): [MA2015] AL2 (9-12) 35 :

35 ) Find inverse functions. [F-BF4]

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. [F-BF4a]

Example: f(x) =2x³ or f(x) = (x+1)/(x-1) for x ≠ 1.

[MA2015] AL2 (9-12) 20 :

20 ) 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-CED1]

[MA2015] AL2 (9-12) 27 :

27 ) 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.* [A-REI11]

[MA2015] AL2 (9-12) 31 :

31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

[MA2015] AL2 (9-12) 36 :

36 ) For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology. [F-LE4]

[MA2019] AL1-19 (9-12) 16 :

16. Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).

[MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

[MA2019] AL1-19 (9-12) 21 :

21. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.

   View Standards     Standard(s): [MA2015] AL2 (9-12) 14 :

14 ) 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.* [A-SSE4]

Example: Calculate mortgage payments.

[MA2015] AL2 (9-12) 31 :

31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

[MA2019] AL1-19 (9-12) 16 :

16. Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).

[MA2019] AL1-19 (9-12) 17 :

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

[MA2019] AL1-19 (9-12) 21 :

21. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.

[MA2019] AL1-19 (9-12) 30 :

30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.