Course Name: Geometry/Math IIUnit 4Unit Title: Functions

Enduring understanding (Big Idea): Students will understand that functions are tools that can be used to interpret real-world phenomena. Students will also understand how to evaluate, graph, and apply properties of linear, quadratic, cubic, exponential, absolute value, and square root functions and their transformations in algebraic and real-world contexts.
Essential Questions:
  1. How are graphs of different functions the same? How are they different?
  2. How can graphs help you understand different real-world situations?
  3. How are functions affected by performing various operations to different aspects of the function?
  4. How do systems of similar and different functions exhibit and apply the properties of all functions?

BY THE END OF THIS UNIT:
Students will know…
  • Formulas from Unit 1 and the formulas and graphs below:
  • Quadratic function: f(x)= ax2 + bx +c
  • Exponential function:
  • Piecewise functions/Step Functions
  • Square root function:
  • Cube root function:
  • Rational function: ,
  • Inverse Variation:
  • Logarithmic function:
  • Absolute value function:
NOTE: Trigonometric Functions will be covered in the Trig Unit.
Vocabulary
Piecewise functions, step functions, greatest integer functions, parent function, transformation, translation, reflection, vertical stretch/compression, absolute value function, axis of symmetry, vertex, parabola, quadratic function, vertex form, max/min value, standard from, power function, constant of proportionality, radical function, square root function, cube root function, inverse variation, constant of variation, rational function, asymptote, slant asymptote, reciprocal function, continuous graph, discontinuous graph, point of discontinuity, removable/non-removable discontinuity, increasing, decreasing, constant, turning point, average rate of change, exponential function, exponential growth and decay, growth/decay factor, logarithm, common log, logarithmic function / Students will be able to…
  • interpret key features of graphs and tables in terms of the quantities and sketch graphs given a verbal description for a function that models a relationship between two quantities (verbal to graph)
  • relate the domain and range of a function to its graph
  • Graph and interpret functions and their transformations (graph to verbal)
  • use factoring to show zeros and symmetry, and interpret in terms of context
  • compare properties of functions represented in a different way (algebraically, graphically, in tables, or by verbal description)
  • write a function that describes a relationship between two quantities
  • identify the effect on the graph of replacing f(x) by f(x)+k, k f(x), f(kx), and f(x+k)
  • Interpret word problems to write functions to model their solutions
  • Compare properties of similar and different functions and their graphs.

Unit Resources:
CCSS-M Included:
F.IF.2, F.IF.4, F.IF.5, F.IF.7b,e, F.IF.8a,F.IF.9, F.BF.1ab, F.BF.3
Test Specification Weights for the
Common Exams in Common Core Math II:
Standard / % Constructed Response / % Multiple-Choice / Category Percentage
(Geometry)
F-IF / 3%-7% / 16% to 19% / 26% to 29%
F-BF / 3% - 7%
/ Mathematical Practices in Focus:
  1. Make sense of problems and persevere in solving.
  2. Reason abstractly.
  3. Model with Mathematics.
  4. Use appropriate tools strategically.
  5. Look for and make use of structure.
  6. Look for and express regularity in repeated reasoning.
Pearson Abbreviation Key:
CC – Common Core Additional Lessons found in the Pearson online materials.
CB- Concept Bytes found in between lessons in the Pearson textbook.
ER – Enrichment worksheets found in teacher resources per chapter.
Puzzle/Activity - these worksheets are also found in the Pearson online materials.
Suggested Pacing:
Intro to Functions
Algebra 1 BK: Section 4.6
Algebra 2 BK: Section 2.1
Function Operations
Algebra 2 BK: Section 6.6 (No composition of functions)
Overview of functions transformations
Algebra 2 BK: 2.6 (ER 2.6/ Activity 2.6)
Linear – quick review
Algebra 1 BK: CC-4, CB 5.3
Exponential – quick review
Algebra 1 BK: CC-11, 7.6,7.7
Algebra 2 BK: 7.1, 7.2, Activity 7.2
Explicit and Recursive Functions
Algebra 1 BK: CC-2, CC-10
Logarithmic Functions
Algebra 2 BK: Section 7.3(pg. 454), CB 7.3
Quadratic
Algebra 1 BK Section 9.1,9.2, CC-13, CC-15,CC-17
Algebra 2 BK Sections 4.1 (ER 4.1/Activity 4.1), 4.2, 4.3, CB 4.3
Absolute Value
Algebra 1 BK: CC-6, Section 5.8, CB 5.8,ER 5.8, Activity 5.8
Algebra 2 BK: Section 2.7 (ER 2.7)
Square/Cube Root
Algebra 1 BK: CC-16, Section 10.5
Algebra 2 BK: Section 6.8 (ER 6.8), CC-4
Polynomial
Algebra 2 BK: Section 5.9 (ER 5.9/Activity 5.9)
Inverse Variation/Rational
Algebra 1 BK: Section 11.6, 11.7, CB 11.7
Algebra 2 BK: CC-6, Section 8.1 (ER 8.1/Activity 8.1), CB 8.2, 8.2 (ER 8.2), 8.3 (ER 8.3/Activity 8.3)
Piecewise/Step
Algebra 1 BK: CC-6
Algebra 2 BK: CB 2.4
Finish
Algebra 2 BK: CC-5

CORE CONTENT

Cluster Title:Understand the concept of a function and use function notation.
Standard: 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.
Note: At this level, extend to quadratic, simple power and inverse variation functions.
Concepts and Skills to Master:
  • Write equations using function notation
  • Use function notation to evaluate functions for given inputs in the domain, including combinations of functions
[(f +g)(x), (fg)(x), (f-g)(x), (f/g)(x) for g(x) ≠0]
  • Interpret statements that use function notation in terms of the context in which they are used

SUPPORTS FOR TEACHERS

Critical Background Knowledge:
  • Evaluate expressions
  • Familiarity with function notation

Academic Vocabulary:
Function notation, evaluate, input, domain, output, range
Suggested Instructional Strategies:
  • Explore a variety of types of situations modeled by functions.
  • Have students create contextual examples that can be modeled by functions.
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
1) Students should continue to use function notation throughout high school mathematics, understanding f(input) = output, f(x)=y.
2) Students should be comfortable finding output given input (i.e. f(3) = ?) and finding inputs given outputs (f(x) = 10) and describe their meanings in the context in which they are used.
Additional note from DPI for Level II:
a. Solve h(5) and explain the meaning of the solution.
b. Solve h(t) = 0 and explain the meaning of the solution.
Notes – All courses are expected to be able evaluate a function given a graphical display. At the course one level, evaluating functions should be limited to linear and exponential. Course two students should extend functions to include quadratic, simple power, and inverse variation functions. / Resources:
Algebra 1 Textbook Correlation:
Section 4.6
Algebra 2 Textbook Correlation:
Section 2.1,6.6
Bank of F.IF.2 specific resources:

Sample Assessment Tasks
Skill-based task:
  1. Given f(x) = 3x find f(4)
  2. What does f(5) = 7 mean?
  3. Write an expression for the relationship depicted
in the graph using function notation. / Problem Task:
Find a function from science, economics, or sports; write it in function notation and explain its meaning at several points in its domain.


CORE CONTENT

Cluster Title: Interpret functions that arise in applications in terms of a context.
Standard: 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.
Note: At this level, trigonometric functions should be limited to simple sine, cosine, and tangent in standard position with angle measures of 180 or less. Periodicity not addressed.
Concepts and Skills to Master:
  • Given a graph, identify key features such as x-and y-intercepts; intervals where the function is increasing, decreasing, positive or negative; symmetry and end-behavior
  • Given a table of values, identify key features such as x- and y-intercepts; intervals where the function is increasing, decreasing, positive or negative; symmetry and end-behavior
  • Find key features of a function and use them to graph the function
  • Use interval notation and symbols of inequality to communicate key features of graphs

SUPPORTS FOR TEACHERS

Critical Background Knowledge:
Ability to graph a linear, exponential, and basic knowledge of quadratic and piecewise-defined function from a table or equationas learned in Algebra 1. Review of linear and exponential functions is recommended before starting quadratic and remaining functions.
Academic Vocabulary:
Increasing, decreasing, positive, negative, intervals, intercepts, interval notation, maximum, minimum, parabola, quadratic function, vertex form, constant of variation, x-intercept, y-intercept, symmetry, end behavior, interval, periodicity, standard form, rational function, asymptote/slant asymptote, reciprocal function, continuous graph, discontinuous graph
Suggested Instructional Strategies:
  • Use graphing technology to explore and identify key features of a function.
  • Use key features of a function to graph functions by hand
  • Ensure that students can identify intercepts, maximum points, minimum points, start and end behavior, symmetry, and periodicity early in the unit.
  • Use technology/graph paper to plot key features and functions.
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
1)Given a function, identify key features in graphs and tables including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
2)This standard should be revisited with every function your class is studying. Students should be able to move fluidly between graphs, tables, words, and symbols and understand the connections between the different representations. For example, when given a table and graph of a function that models a real-life situation, explain how the table relates to the graph and vice versa. Also explain the meaning of the characteristics of the graph and table in the context of the problem as follows:
At the course two level, students should extend the previous course work with function types to focus on power functions, and inverse functions.
  • Power functions – the effects of a positive/negative coefficient, the effects of the exponent on end behavior, the rate of increase or decrease on certain intervals,
  • Inverse functions – understanding the effects on the graph of having a variable in the denominator (asymptotes).
(View DPI documents for more detail on difference between Math I and Math III.)
3)When given a verbal description of the relationship between two quantities, sketch a graph of the relationship, showing key features.
Note – This standard should be seen as related to F-IF.7 with the key difference being students can interpret from a graph or sketch graphs from a verbal description of key features. / Starting Resources:
Algebra 1Textbook Correlation:
Sections 4.2, 4.3, ,9.1, 9.2, 9.7, 11.7
Algebra 2 Textbook Correlation:
Sections 2.1, 2.6, 4.1-4.3, CB4.3, 5.1,5.8,CB 7.3, CC-3, CC-5,5.9, 6.8, 7.1, 7.2, 13.1, 13.4, CB 13.4, 13.5, 13.6
MARS Lesson: Interpreting Algebraic Functions
Bank of F.IF.4 specific resources:

Sample Assessment Tasks
Skill-based task:
  1. Identify the intervals where the function is increasing and decreasing.

  1. On what interval does the graph of f(x) = x2 increase?
/ Problem Task:
Given the graph of a function, identify the intercepts, minimum and maximum values, lines of symmetry, and end behavior. Using the basic functions learned in the previous objective how is this function similar/different?
Additional note from DPI for Level II:
Ex. Insert graph or equation of a polynomial function from a real-life situation.
a. What are the x-intercepts and y-intercepts and explain them in the context of the problem?
b. Identify any maximums or minimums and explain their meaning in the context of the problem.
c. Describe the intervals of increase and decrease and explain them in the context of the problem.

CORE CONTENT

Cluster Title: Interpret functions that arise in applications in terms of a context.
Standard: F.IF.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.
Note: At this level, extend to quadratic, right triangle trigonometry, and inverse variation functions.
Concepts and Skills to Master:
  • Determine the domain of a function from its graph, including analysis of the start and end behavior.
  • Determine the key domain restrictions of functions, including division by 0 and even roots of negative numbers.
  • Analyze word problems to determine domain restrictions (i.e. negative numbers, non-integers).

SUPPORTS FOR TEACHERS

Critical Background Knowledge:
  • Familiarity with function notation and domain
  • Knowledge of independent and dependent variables

Academic Vocabulary:
Domain, function, integers, independent/dependent variable, restricted domain, Start Behavior, End Behavior, parabola, quadratic function, vertex form, min/max value, constant of variation
Suggested Instructional Strategies:
  • Discuss contexts where the domain of a function should be limited to a subset of integers, positive or negative values, or some other restriction to the real numbers
  • Find examples of functions with limited domains from other curricular areas (science, physical education, social studies, consumer science…)
  • Compare the graphs of parent functions to their equations to determine any restrictions algebraically.
  • Identify the range of functions based on their graphs and equations.
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
Given a function, determine its domain. Describe the connections between the domain and the graph of the function. Know that the domain taken out of context is a theoretical domain and that the practical domain of a function is found based on a contextual situation given, and is the input values that make sense to the constraints of the problem context. / Starting Resources:
Algebra 1 Textbook Correlation:
Sections 4.4,9.1, 11.6
Algebra 2 Textbook Correlation:
Sections 2.1, 4.3, 5.8
Bank of F.IF.5 specific resources:

Sample Assessment Tasks
Skill-based task:
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.
What are the domains of f(x) = x, f(x) = x2, f(x) = x3,
f(x) = √x, and f(x) = 1/x? Explain the rationale for any restrictions (or lack of restrictions). / Problem Task:
Create functions in context where the domain would be:
  1. All real numbers
  2. Integers
  3. Negative integers
  4. Rational numbers
  5. (10, 40)
The summer before going to college, a student earned a promotion to shift supervisor at her job at Starbucks! The new position pays $10.20/hour. If the student’s paycheck was modeled by a function, what would be the domain and range? Explain your answer. Why is this domain and range different than other similar functions?
NCDPI Examples:
Ex. A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h(t) = -16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet.
a. What is the theoretical domain for the function? How do you know this?
b. What is the practical domain for t in this context? Explain.
c. What is the height of the rocket two seconds after it was launched?
d. What is the maximum value of the function and what does it mean in context?
e. When is the rocket 100 feet above the ground?
f. When is the rocket 250 feet above the ground?
g. Why are there two answers to part e but only one practical answer for part d?
h. What are the intercepts of this function? What do they mean in the context of this problem?
i. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the problem?

CORE CONTENT

Cluster Title: Analyze functions using different representations
Standard: 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.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and graph trigonometric functions, showing period, midline, and amplitude.
Note: At this level, extend to simple trigonometric functions (sine, cosine, and tangent in standard position.)
Concepts and Skills to Master:
  • Graph parabolas expressed in vertex form or factored form by hand.
  • Graph piecewise-defined functions by hand
  • Use technology to model complex square root and piecewise-defined functions
  • Identify intercepts, maxima, minima, end behavior, increasing/decreasing intervals, and axis of symmetry in graphs of quadratic functions
  • Identify, compare and analyze, the key features (domain, range, intercepts, transformations) of exponential and logarithmic functions.
FOR e: Given real-world data, identify behaviors that tend to be explained using logarithmic and exponential functions.

SUPPORTS FOR TEACHERS

Critical Background Knowledge:
Graph points on the coordinate plane
Processes for graphing functions in a calculator:
  • Locating and identifying parent functions, intercepts, minimum, and maximum values, graphing points, adjust calculator windows to appropriately model data
  • Evaluating Functions

Academic Vocabulary:
Quadratic function, parabola, vertex form, piecewise-defined, step function, intercept, end behavior, increasing, decreasing, maximum value, minimum value, radical function, square root function, cube root function, inverse variation, Absolute value function, axis of symmetry, vertex, x-intercept, y-intercept, Base, logarithm function, common log, exponential function, exponential growth and decay
Suggested Instructional Strategies:
  • Allow students to develop graphs from tables and use those graphs to generalize graphing strategies.
  • Graph equations generated from real-life contexts
  • Compare equations, tables, and graphs to find intercepts, minimum, and maximum values of functions.
  • Use real-world situations to find key points of functions and interpret them in a real-world context.
  • Demonstrate graphing calculator processes for complex functions.
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
Part a., b., and c. are learned by students sequentially in Courses I – III.
Part e. is carried through Courses I – III with a focus on exponential in Course I and moving towards logarithms in Courses II and III.
This standard should be seen as related to F-IF.4 with the key difference being students can create graphs, by hand and using technology, from the symbolic function in this standard.
In Algebra I and Mathematics II, for F.IF.7b, compare and contrast absolute value, step and piecewise- defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise- defined functions. / Starting Resources:
Algebra 1 and Algebra 2 Textbook Correlation:
See Part II of suggested pacing
Geogebra (free online)
Bank of F.IF.7b specific resources:

Bank of F.IF.7e specific resources:

Sample Assessment Tasks
Skill-based task:
1)Graph the function f(x) = 3|2x + 1| - 2 and identify any maxima or minima.
2)Graph the function y = (x – 3)(x + 2), identify the axis of symmetry and the vertex
3)Graph in the calculator and label all key points:
f(x) = ½(x + 3)2 + 2 / Problem Task:
Jose created a distance vs time graph by starting 2 meters away from the wall. He walked towards to wall at 0.25 m/sec for 4 seconds, stood still for 2 seconds, walked away from the wall at 0.4 m/sec for 2 seconds, and then stopped for 2 seconds. Sketch Jose’s distance vs time graph. What was Jose’s ending position? Give both the distance from the wall and the time.
Two baseball players tracked the paths of their most recent home runs using quadratic functions. The height of one player’s baseball modeled the function h(t) = -16(t – 4)2 + 260, and the height of the second player’s baseball modeled the function h(t) = -16(t – 4)2 + 258. What are the similarities and differences between the graphs? What do those similarities and differences represent in a real-world context?
NCDPI Examples:
Ex. Income taxes are calculated at a rate dependent on the range of a person’s taxable income. The table below shows income tax rates up to 100,000.
Taxable Income (dollars) Income Tax
$0-$10,000 10% of income
$10,001-$45,550 15% of income over 10,000 plus $1,000
$45,551-$100,000 25% of income over 45,552 plus $6832.50
a. Last year, Ravi earned $15,000 washing cars. How much should he pay in personal income taxes?
b. Jordan earned $50,000 managing a car washing business. How much does he owe in taxes?
c. Explain the connections between this piecewise function and the tax table.

d. Graph this function and label any characteristic features that are key to the function.

CORE CONTENT