Curriculum  /  Math  /  9th Grade  /  Unit 5: Functions and Transformations  /  Lesson 15

Functions and Transformations

Lesson 15

Math

Unit 5

9th Grade

Lesson 15 of 16

Objective

Identify and describe horizontal scaling of functions, including reflections over the $$y$$-axis.

Common Core Standards

Core Standards

  • F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  • 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.

Foundational Standards

  • 8.F.B.4
  • 8.G.A.3

Criteria for Success

  1. Identify when the graph of a function has been scaled horizontally, both in a graph and in an equation.
  2. Describe how to scale a function’s graph horizontally in a table of values or graphically.
  3. Draw graphs of functions that have been scaled horizontally.
  4. Write equations, in function form, to represent graphs of functions that have been scaled horizontally using $$b$$ to represent the vertical scale (i.e., $$f(bx)$$).
  5. Understand that when $$b$$ is negative, then the graph of the function is reflected over the $$y$$-axis.
  6. Compare vertical scaling to horizontal scaling. 
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Anchor Problems

Problem 1

Consider the two functions below. 

$${f(x)=|x|}$$

$${g(x)=|2x|}$$

  1. How do you think the graphs of the two functions will be different? How will they be the same? 
  2. Look at the corresponding work in the Desmos activity, Introduction to Transformations of Functions, slides 2–5. 
  3. Describe how the function $${h(x)=|bx|}$$ changes as the value of $$b$$ changes. Refer to the Desmos activity, slide 10.

Guiding Questions

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References

Desmos Introduction to Transformations of Functions — Slides 2-5 and 10

Introduction to Transformations of Functions by is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 10, 2018, 4:26 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Consider the two functions below.

$${g(x)=2x^2}$$

$${h(x)=(2x)^2}$$

  1. Sketch a graph of the two functions in the same coordinate plane. Include also a sketch of the parent function, $${f(x)=x^2}$$.
  2. Which function represents a vertical scaling of the graph of $${{ f(x)}}$$?
  3. Which function represents a horizontal scaling of the graph of $${{ f(x)}}$$?
  4. Compare and contrast functions $$g$$ and $$h$$.

Guiding Questions

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Problem 3

Consider the two functions below.

$${g(x)=-\sqrt{x}}$$

$${h(x)=\sqrt{-x}}$$

  1. Sketch a graph of the two functions in the same coordinate plane. Include also a sketch of the parent function, $${f(x)=\sqrt{x}}$$.
  2. Compare and contrast functions $$g$$ and $$h$$.

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

Let $${f(x)=x^2}$$, $${g(x)=(3x)^2}$$, and $${h(x)=({1\over3}x)^2}$$, where $$x$$ can be any real number. The graphs below are of $$y=f(x)$$, $$y=g(x)$$, and $$y=h(x)$$. 

  1. Label each graph with the appropriate equation. 
  2. Describe the transformation that takes the graph of $$ y=f(x)$$ to the graph of $$ y=g(x)$$. Use coordinates of each to illustrate an example of the correspondence. 
  3. Describe the transformation that takes the graph of $$y=f(x)$$ to the graph of $$y=h(x)$$. Use coordinates to illustrate an example of the correspondence. 

References

EngageNY Mathematics Algebra I > Module 3 > Topic C > Lesson 19 — Exit Ticket

Algebra I > Module 3 > Topic C > Lesson 19 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.

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Lesson 14

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Lesson 16

Lesson Map

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

Topic A: Piecewise Functions

Topic B: Absolute Value Functions

Topic C: Function Transformations

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