Functions, Graphs and Features

Lesson 2

Math

Unit 1

9th Grade

Lesson 2 of 11

Objective


Define functions and evaluate points in tables, graphs, and contextual situations using function notation.

Common Core Standards


Core Standards

  • F.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • 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.

Foundational Standards

  • 8.F.A.1

Criteria for Success


  1. Define functions as relationships where each input has only one output in contextual and non-contextual situations represented in tables and graphs. 
  2. Use function notation to describe a function of a situation by denoting the function f. Describe the input as $$x$$, the output as f($$x$$), and a coordinate point in function notation (e.g., f(3) = 5 is the coordinate point (3, 5)).
  3. Evaluate a function represented graphically given in function notation.
  4. Use appropriate language to represent functions ("f of $$x$$"), and describe a situation in terms of independent variables (i.e., "Time is a function of distance." Represent this as function notation where t is the time, and f(t) represents the distance.)

Tips for Teachers


  • Students may need to review the definition of functions, as represented in the standard 8.F.1, specifically graphically and in a table, before they can fully access this lesson. 
  • A misconception students may have is to treat function notation as multiplication. Use the oral language of “f of x” and name this misconception to students to help alleviate confusion. 
  • This is the first lesson that introduces function notation to students. We are specifically focusing on naming functions and identifying/writing points, inputs, and outputs in function notation. Using function notation to define equations that represent a function will come later. 
  • The following resource may be helpful to see all the ways function notation can be represented.
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Anchor Problems

25-30 minutes


Problem 1

Below is an example of a relation that is a function and an example of a relation that is not a function

What are the defining characteristics of a function? 

FUNCTION:

$$x$$ $$y$$
3 6
4 6
5 3
7 8

NOT A FUNCTION:

$$x$$ $$y$$
3 5
4 6
4 3
7 8

 

Guiding Questions

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Problem 2

Use the graph below to answer the questions that follow: 

Function notation describes a particular relationship and implies the dependency of variables. 

The general form of function notation is $${f(x) = y}$$.  $$y$$ is dependent on $${x }$$.

Part A: 
The function $$f$$ above has a point $$f(-2) = 1$$. Mark this point on the graph above. 

Part B: 
Find the $$x$$ or $$y$$ values from the graph noted by the function notation below. 

$$f(-9)=$$

$$f(x)=1$$

$$f(x)=-2$$

$$f(4)=$$

$$g(x)=5$$

$${g(0)=}$$

Guiding Questions

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Problem 3

John was running a race, and his coach was timing him. John’s race can be described as the distance in meters, $$d$$, as a function of time in seconds, $$t$$.  Partway through the race, John’s position could be described as $$d(6)=50$$. Describe John’s time and distance at this point. 

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

Suppose $$f$$ is a function. 

  1. If $$10 = f(-4)$$, give the coordinates of a point on the graph of $$f$$
  2. If 6 is a solution to the equation $$f(w) = 1$$, give a point on the graph of $$f$$

References

Illustrative Mathematics Points on a Graph

Points on a Graph, accessed on June 22, 2017, 3:51 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.

Problem 2

Let $${f(t)}$$ be the number of people, in millions, who own cell phones $$t$$ years after 1990. Explain the meaning of the following statements. 

  1. $${f(10)=100.3}$$
  2. $${f(a)=20}$$
  3. $${f(20)=b}$$ 
  4. $$n= {f(t)}$$

References

Illustrative Mathematics Cell Phones

Cell Phones, accessed on June 22, 2017, 3:53 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.

  • Reason and Wonder, “The Date Game,” blog entry by Michael Fenton.
  • Sample problem: When does $${{f(x)} = {g(x)}}$$?  What is $${f(x)}$$ at this point? What is $${g(x)}$$ at this point? What is $${f(-6)}$$?  What is the solution of $${g(x)} = 1$$

  • Illustrative Mathematics Yam in the OvenInclude this as a follow-up to this problem: Write this statement in function notation: “The temperature of the yam after being in the oven for 30 minutes is 120 degrees.
  • Illustrative Mathematics The Parking Lot
  • Mathematics Vision Project: Secondary Mathematics One Module 5: Systems of Equations and InequalitiesLesson 5.4
  • Mathematics Vision Project: Secondary Mathematics One Module 3: Features of FunctionsLesson 3.7: To Function or Not to Function - A Practice Understanding Task

Next

Identify features of functions, including x-intercept and y-intercept, in context. Evaluate function notation in context.

Lesson 3
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Lesson Map

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

Topic A: Features of Functions

Topic B: Nonlinear Functions

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