The Mathematics of Functions
Unit Plan
The Mathematics of Functions
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Grade Levels
- Related Academic Standards
- Assessment Anchors
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Eligible Content
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Big Ideas
- Bivariate data can be modeled with mathematical functions that approximate the data well and help us make predictions based on the data.
- 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 (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
- Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
- Patterns exhibit relationships that can be extended, described, and generalized.
- Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
- There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
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Concepts
- Algebraic properties and processes
- Algebraic properties, processes and representations
- Analysis of one and two variable (univariate and bivariate) data
- Exponential functions and equations
- Functions and multiple representations
- Linear relationships: Equation and inequalities in one and two variables
- Linear system of equations and inequalities
- Polynomial functions and equations
- Quadratic functions and equations
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Competencies
- Extend algebraic properties and processes to quadratic, exponential, and polynomial expressions and equations and to matrices, and apply them to solve real world problems.
- Represent a polynomial function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated polynomial equation to each representation.
- Represent a quadratic function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated quadratic equation to each representation.
- Represent exponential functions in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the growth/decay rate of the associated exponential equation to each representation.
- Represent functions (linear and non-linear) in multiple ways, including tables, algebraic rules, graphs, and contextual situations and make connections among these representations. Choose the appropriate functional representation to model a real world situation and solve problems relating to that situation.
- Use algebraic properties and processes in mathematical situations and apply them to solve real world problems.
- Write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques.
- Write, solve, graph, and interpret linear equations and inequalities to model relationships between quantities.
Objectives
This unit explores functions, from the definition and various representations of functions through the concepts of inverse functions, as well as operations with functions, specifically composition. Students will:
- learn the definition of a function and distinguish functions from nonfunctions.
- learn to find the inverse of functions algebraically using a table or graph.
- compose functions written as equations as well as graphs.
Essential Questions
- What is a function and why are functions so mathematically interesting and necessary?
Related Unit and Lesson Plans
Related Materials & Resources
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Formative Assessment
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View
Multiple-Choice Items:
1. Which of the following graphs is not a function?
2. Based on this table, which of the following statements is true?
x
y
1
3
2
1
3
5
A f(3) = 7
B f(1) = 2
C f(1) = 3
D f(2) does not exist
3. A doctor knows that medicine is most effective immediately after a patient takes it. Then the effectiveness of the medicine decreases. After 6 hours, the doctor gives the patient more medicine. Which graph could model this situation?
4. Lulu is sewing a dress. The dress needs 2 feet of ribbon around the waist. Also, Lulu will add bows, each of which requires 0.5 foot of ribbon. Which table correctly models the situation?
A
Number of Bows
Total Feet of Ribbon Needed
1
3
2
5
3
7
B
Number of Bows
Total Feet of Ribbon Needed
1
2
2
2
3
2
C
Number of Bows
Total Feet of Ribbon Needed
1
2
2
2.5
3
3
D
Number of Bows
Total Feet of Ribbon Needed
1
2.5
2
3
3
3.5
Use the following graph to answer question 5.
5. The graph above models Maria’s bike ride to school. Which situation describes her ride most accurately?
A Maria left home, rode for 2 minutes, and then returned home to get her homework. After getting her homework, she finished her bike ride to school.
B Maria left school, rode for 3 minutes, rode back to school, and then rode home.
C Maria rode straight to school. The entire ride took 2 minutes.
D Maria rode her bike for 3 minutes and then stopped to rest for 2 minutes. Then she rode the remaining distance.
6. Which of these tables is not a function?
A
x
y
2
3
3
3
4
3
B
x
y
1
5
1
4
1
3
C
x
y
−2
4
0
0
2
4
D
x
y
5.1
4.2
3.9
5.8
2.7
6.1
Use the graph below to answer question 7.
7. Which of the following tables represents this graph?
A
x
f(x)
−1
3
0
4
1
5
B
x
f(x)
−1
5
0
4
1
3
C
x
f(x)
3
−1
4
0
5
1
D
x
f(x)
−1
−1.5
0
1
1
1.5
8. How are the graph of a function and the graph of its inverse related?
A They are reflections of one another across the x-axis.
B They are reflections of one another across the y-axis.
C They are reflections of one another across the line y = x.
D They are reflections of one another across the origin.
9. For the functions f(x) = 2x − 5 and g(x) = 3x − 2, what is f(g(x))?
A. 2x − 5
B. 6x − 9
C. 4x − 15
D. 6x − 17
Multiple-Choice Answer Key:
1. C
2. C
3. B
4. D
5. D
6. B
7. B
8. C
9. B
Short-Answer Items:
10. Find the inverse of and draw its graph.
11. Let f(x) = 2x − 1 and g(x) = x2. Find f(g(2)).
12. Graph . Is this a function? Using the vertical line test, tell why it is or is not a function.
Short-Answer Key and Scoring Rubrics:
Remember that f-1(x) means the inverse function of f(x), not f to the negative 1 power. Students may use another letter to represent the inverse function as well, so be prepared for that.
10. Find the inverse of and draw its graph.
The graph for the inverse should look like this, though keep in mind scales and viewing regions may vary:
Points
Description
2
- Response is complete, correct, and detailed. Student translates between the function and its inverse. The graph is correctly drawn with slope =2 and y-intercept = 6.
1
- Response is partially correct, but has no graph or the graph does not represent the inverse.
0
- Incorrect response; no graph or incorrect graph.
11. Let f(x) = 2x − 1 and g(x) = x2. Find f(g(2)).
f(g(x)) = 2(2)2−1 = 7
Points
Description
2
- Response is complete and correct, showing the calculation.
1
- Response is correct, but includes no supporting calculations or path to the solution.
0
- Incorrect response; no appropriate supporting work.
12. Graph . Is this a function? Using the vertical line test, tell why it is or is not a function.
Yes. The relation is a function. The vertical line test shows one and only one output for each input.
Points
Description
2
- Response is Yes and indicates that the vertical line test shows one and only one output for each input.
1
- Response is Yes, but includes no adequate supporting work or path to the solution.
0
- Incorrect response; no appropriate supporting work.
Performance Assessment:
A certain cell phone company charges $0.03 per text message as well as a flat rate of $4 per month. Show your work.
- Let T stand for the number of text messages sent and C stand for total cost. Write a linear equation to model this situation where C is a function of T.
- Find the inverse of your function from Part 1.
- Explain what the inverse means in terms of the situation.
Performance Assessment Scoring Rubric:
- C = 4 + 0.03 * T
- The inverse tells you how many texts you can send for a certain monthly cost.
Points
Description
4
- Both equations are correctly represented in terms of C and T.
- Inverse is expressed as number of texts per cost.
- Supporting work is complete and correct.
3
- Parts 1 and 2, or 2 and 3, or 1 and 3 are correct.
- Appropriate supporting work is provided for Part 1 or 2.
2
- Part 1 or 2 or 3 is correct.
- Appropriate supporting work that leads to the solution is provided.
1
- Part 1 or 2 or 3 is correct.
- Supporting work is not present.
0
- Parts 1, 2, and 3 are incorrect.
- Supporting work is incorrect, unclear, or not provided.