Linear Functions and Applications

Lesson 2

Math

Unit 1

11th Grade

Lesson 2 of 13

Objective


Write linear functions that represent contextual situations.

Common Core Standards


Core Standards

  • 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.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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • 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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

  • 8.F.A.1
  • 8.F.B.4

Criteria for Success


  1. Identify dependency in a function and use function notation to illustrate that dependency. 
  2. Identify domain restrictions based on the context. 
  3. Sketch functions on a coordinate plane according to the constraints on variables. 
  4. Identify the features of a linear function graphically, algebraically, and in context. 
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Anchor Problems


Problem 1

Below is a list of quantities and associated variables that can be used to describe a situation. 

  • A large swimming pool has a capacity of $$y$$ gallons. 
  • The pool has already been filled with $$w$$ gallons. 
  • After $${t_{1}}$$ minutes, the pool is filled with $${g_{1}}$$ gallons. 
  • After $${t_{2}}$$ minutes, the pool is filled with $${g_{2}}$$ gallons. 
  • The pool will take $$r$$ minutes to completely fill to $$y$$ gallons. 
  1. What is the independent variable? What is the dependent variable? 
  2. What is the meaning of each of the expressions below?

Guiding Questions

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Problem 2

The function from the situation above is written in different variables, where the amount the pool has filled is a function of the number of minutes that have passed. 

$${f(x) = mx + 4}$$

  1. You know that $${f(3) = 5.}$$ What is the rate at which the pool is filling?
  2. What are the dependent and independent variables in this situation? 

Guiding Questions

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Target Task


Problem 1

Below is a graph of the amount of gas left in the tank as a function of the amount of miles traveled. Write a function in function notation that describes this situation. Label the axes with the appropriate units.

Problem 2

The school librarian is unpacking books that he has ordered. The number of books he can unpack and shelve, $${b(t)}$$ can be expressed as a function of time in minutes, t, and is a proportional relationship. What does each of the following mean in the context of the problem? 

$${b(5) = 20}$$

$${b(t)} = 32$$

$${b(20)}$$

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include an extension to Anchor Problem #1: “Assuming a linear function, write a model that describes the amount of total gallons in the pool as a function of the amount of time that has passed.”
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Lesson 1

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Lesson 3

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Linear Functions

Topic B: Systems of Functions and Constraints

Topic C: Piecewise Functions

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