Functions, Graphs and Features

Lesson 5

Math

Unit 1

9th Grade

Lesson 5 of 11

Objective


Calculate and interpret the rate of change from two points on a graph, in a situation, or in function notation.

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.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 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.EE.B.5
  • 8.F.A.3

Criteria for Success


  1. Describe the average rate of change as how much the range of an interval changes relative to the domain of an interval. 
  2. Find the rate of change between two endpoints of an interval to represent the average rate of change over that interval. 
  3. Represent intervals in an inequality showing the range of domain represented, two points given in function notation, coordinate points, or rows in a table of values. 
  4. Describe that the rate of change of a linear function is always constant.
  5. Use the rates of change over different intervals to interpret a function.

Tips for Teachers


Students may need to review the slope formula from Grade 8 before they can fully access this lesson. 

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Anchor Problems


Problem 1

Given the graph below, over which interval is the average rate of change the greatest? Explain your reasoning. 

a. Between $${f(4)}$$ and $${f(10)}$$
b.  $${10 \leq x \leq 15}$$
c. Between $${(0,0)}$$ and $${(4,11)}$$
d. From $${x=0}$$ to  $${x=10}$$

Guiding Questions

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Problem 2

The table below defines the relationship $${y=f(x)}$$

Find the average rate of change from $${f (0)}$$ to $${f(5)}$$ . 

$$x$$ 0 0 4 5
$$f(x)$$ 26 17 5 1

 

Guiding Questions

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Problem 3

You leave from your house at 12:00 p.m. and arrive to your grandmother’s house at 2:30 p.m. Your grandmother lives 100 miles away from your house. 

What was your average speed over the entire trip from your house to your grandmother’s house? 

Guiding Questions

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Problem Set


Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task


The table below shows the temperature, $$T$$, in Tucson, Arizona, $$t$$ hours after midnight. 

$$t$$ 0 3 4
$$T$$ 85 76 70

When does the temperature decrease the fastest, between midnight and 3 a.m. or between 3 a.m. and 4 a.m.? Explain your reasoning. 

References

Illustrative Mathematics Temperature Change

Temperature Change, accessed on June 22, 2017, 1:24 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

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.

  • As an extension, include a problem or two where students need to sketch a function that has a particular rate of change. 
  • As an extension, include a problem where students need to find the average rate of change over an interval in a function presented algebraically. For example: Find the rate of change from  $${x = 2}$$  to $${x = -1}$$ with respect to the linear equation $${2y +3x = 12}$$.
  • Include problems where students are finding the average rate of change from nonlinear graphs. 
  • Include problems where students need to find the average rate of change from a table of values. For example: 

Find the average rate of change between the named intervals in the table of values. 

 

 

x $${f(x)}$$
-3 1
0 2
3 3
6 4
9 5

Between $${f(-3)}$$ and $${f(3)}$$

Between $${{f(0)}}$$ and $${f(6)}$$

Between $${{f(0)}}$$ and $${{f(9)}}$$

Between $${f(3)} $$ and $${{f(9)}}$$

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Lesson 4

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Lesson 6

Lesson Map

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

Topic A: Features of Functions

Topic B: Nonlinear Functions

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