Curriculum  /  Math  /  9th Grade  /  Unit 1: Functions, Graphs and Features  /  Lesson 10

Functions, Graphs and Features

Lesson 10

Math

Unit 1

9th Grade

Lesson 10 of 11

Objective

Draw a graph to represent a system of functions. Identify the solution to a system represented graphically and in context.

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.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.
  • A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 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.C.8
  • 8.F.B.5

Criteria for Success

  1. Describe a system of functions as any set of functions that happen over the same domain.
  2. Describe that the solution to a system of functions is when the functions intersect. 
  3. Describe the intersection of the where both functions have the SAME coordinate point (same x-coordinate, same y-coordinate). For example, in a distance/time graph, the intersection point is where both the distance and the time is the same for both functions. 
  4. Describe the solution to a system of functions in context.

Tips for Teachers

This lesson asks students to look at a system of functions with a graphical analysis lens rather than a solution lens. Students will need to be able to describe the system in the context of the problem and identify that the intersection of the functions is where both functions have the same value (in context). Significantly more work on systems of linear equations and functions will be started in Unit 4 and then continued in Units 5, 6, 7, and 8. 

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Anchor Problems

25-30 minutes

Problem 1

In this video, a man and a girl are walking up and down a set of stairs. Draw a graph that models each of their elevations as a function of time. 

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 5 — Example 1: Man and Girl on Stairway

Algebra I > Module 1 > Topic A > Lesson 5 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Below are three people eating popcorn at different rates. 

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Mr. Orr Is a Geek.com Popcorn Pandemonium

Popcorn Pandemonium by Jon Orr is made available on Mr. Orr is a Geek.com under the CC BY 4.0 license. Accessed June 23, 2017, 10:56 a.m..

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

Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 feet apart. Each person starts at his or her own door and walks at a steady pace toward the other. They stop walking when they meet. 

Suppose: 

  • Maya walks at a constant rate of 3 feet every second. 
  • Earl walks at a constant rate of 4 feet every second. 
  1. Graph both people’s distance from Maya’s door versus time in seconds. 
  2. According to your graphs, approximately how far will they be from Maya’s door when they meet? 

References

EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 5 — Exit Ticket

Algebra I > Module 1 > Topic A > Lesson 5 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

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.

  • Provide problems for students where there is just one function to review, and then add an additional function. 
  • Give students a graph of a system of functions along with variables and quantities on the graph. Ask students to explain the features of the system with respect to domain, range, linearity, rate of change, and intercepts.

Next

Analyze functions and identify parent functions of graphs. Identify variables of a situation and the scale of the associated graph. Represent a situation in a graph, table, and description.

Lesson 11
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Lesson Map

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

Topic A: Features of Functions

Topic B: Nonlinear Functions

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