Curriculum  /  Math  /  9th Grade  /  Unit 1: Functions, Graphs and Features  /  Lesson 6

Functions, Graphs and Features

Lesson 6

Math

Unit 1

9th Grade

Lesson 6 of 11

Objective

Describe and sketch functions using the features of domain and range, intercepts, function behavior, and the value of the function.

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.

Foundational Standards

  • 8.F.B.5

Criteria for Success

  1. Describe the intervals of a function according to the behavior of the function (increasing, decreasing, constant). 
  2. Describe the intervals of a function according to the value of the function (positive, negative, zero). 
  3. Describe the intervals of a function according to important points and the $$x$$ and $$y$$ intercepts.
  4. Use inequalities to describe the intervals where a function meets certain criteria. 
  5. Draw a sketch of a function according to particular features. 

Tips for Teachers

This lesson marks a very important introduction to function modeling—the idea of a sketch of a graph. Teachers should consider the level of precision that is expected from students with sketching a graph in this lesson in order to understand what is happening with a function. For further discussion on this topic, see Dan Meyer, “Developing the Question: Ask for a Sketch First.”

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Anchor Problems

25-30 minutes

Problem 1

Describe the function below so that someone could recreate the shape from your description. 

Guiding Questions

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Problem 2

Sketch a function that meets the following requirements.

  • Domain $${-4 \leq x < 6}$$
  • Range $${-3 \leq y < 4}$$
  • Increasing at a constant rate over the interval $${-4 \leq x < -1}$$
  • Decreasing at a constant rate over the interval $${-1< x \leq 2}$$
  • x-intercept at $${x=-2}$$
  • y-intercept at $${y=2}$$

Guiding Questions

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Problem Set

15-20 minutes

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

Target Task

5-10 minutes

Problem 1

Describe this graph in terms of: 

  • Domain/range
  • Function behavior
  • Intercepts


Provide specific evidence for where the function is increasing, decreasing, or constant, using rate-of-change calculations. 

Problem 2

Sketch a graph that has the following requirements: 

  • Decreasing at a constant rate of change over the interval  $${0 \leq x \leq 5}$$ 
  •  x-intercept at  $${x = 5}$$
  • Increasing from $${f(5)}$$ to  $${f(8)}$$
  • Range of   $${0 \leq y \leq 6}$$ 

Based on these criteria, what is a possible domain of this function? 

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.

  • Alternative Math, “Time Distance Graphs,” blog post, has a gallery walk/station activity that could easily be turned into practice problems.
  • Identify key features of graphs. Give students problems on a function that is positive versus increasing, a function that is negative versus decreasing, and how this is different in both function notation and in the graph itself. 
  • RDA Performance Task Bank SBAC Item MAT.HS.CR.1.00FIF.L.614
  • MARS Formative Assessment Lessons for High School Representing Functions of Everyday Situations — This gets into contextual sketching but will lead students into tomorrow’s lesson. The resource has matching cards of the situation, the graph, and the function. I suggest not matching to the equation; many of the functions are beyond what they will study

Next

Analyze the key features of a contextual situation and model these graphically.

Lesson 7
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Lesson Map

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

Topic A: Features of Functions

Topic B: Nonlinear Functions

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