Linear Functions and Applications

Lesson 1

Math

Unit 1

11th Grade

Lesson 1 of 13

Objective


Identify features of linear functions from equations, verbal descriptions, tables, and graphs.

Common Core Standards


Core Standards

  • 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.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.
  • A.SSE.A.1.A — Interpret parts of an expression, such as terms, factors, and coefficients.

Foundational Standards

  • 8.F.A.2
  • A.REI.B.3

Criteria for Success


  1. Write a linear equation in slope-intercept, standard, and point-slope form to reveal features of functions. 
  2. Identify features of a linear function, such as y-intercept, slope, x-intercept, and increasing/decreasing, from a table of values, equation, and graphs. 
  3. Graph linear functions from equations by defining a table of values. 
  4. Compare features of linear functions represented in different forms. 

Tips for Teachers


  • This lesson is an eighth-grade/Algebra 1 reintroduction to the features of a linear equation, manipulating a linear equation, and graphing a linear equation. If students are proficient at these skills, this lesson can be skipped. 
  • Students may need to review solving for a variable before they can fully access this lesson. If students are not proficient in this skill, the target task can be adapted so that students are not asked to find the largest possible y-intercept but just to fill in values for a, b, and c and then graph or make a table of values. 
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Anchor Problems


Problem 1

All of the four quadrants represent the same function but use a different representation. Explain how you know that each of these quadrants represents the same function. 

Guiding Questions

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Problem 2

Describe a characteristic that each of the quadrants shown below does not share with the rest. 

You should have at least one characteristic for each quadrant. 

Guiding Questions

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References

Which One Doesn't Belong Graphs & EquationsGraph 8 from Erick Lee

Graphs & Equations is made available on Which One Doesn't Belong?. Copyright © 2013 wwdb.ca. All Rights Reserved. Accessed on Sept. 26, 2017, 2:54 p.m..

Problem 3

Allison states that the slope of the following equation is 3. 

$${3x + 4y = 8}$$

Allison creates a table of values for the equation and is confused when it does not appear that the slope in the table is 3. 

  • Explain how Allison found the slope from the table. 
  • Explain the mistake Allison made in finding the slope from the equation. 
  • Graph the equation. 

Guiding Questions

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Target Task


Assign values to $${a, b,}$$ and $$c$$  with the numbers -9 to 9, such that an equation with the largest possible y-intercept is formed. 

$$ax + by = c$$

 

a.   Write a table of values for this equation over the domain.

b.   Graph this equation.

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 problems like: 
    • Given two or three different equations, how do you know that each of the equations are equivalent? For example: 

Equation #1: $${y = 3x + 7}$$

Equation #2: $${(y - 4) = 3(x + 1)}$$

Equation #3: $${2y - 6x = 14}$$

  • Given the situation, choose an equation that models the situation. 
  • Given the graph, choose the equation that describes the graph. 
  • Given the equation, choose the graph that models the equation. 
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Lesson 2

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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