Curriculum  /  Math  /  9th Grade  /  Unit 5: Functions and Transformations  /  Lesson 14

Functions and Transformations

Lesson 14

Math

Unit 5

9th Grade

Lesson 14 of 16

Objective

Identify and describe vertical scaling of functions, including reflections over the $$x$$-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 vertically, both in a graph and in an equation.
  2. Describe how to scale a function’s graph vertically in a table of values or graphically.
  3. Draw graphs of functions that have been scaled vertically.
  4. Write equations, in function form, to represent graphs of functions that have been scaled vertically using a to represent the vertical scale (i.e., $${af(x)}$$).
  5. Differentiate between vertical stretches, where $${|a|>1}$$, and vertical shrinking, where $${ |a|<1}$$.
  6. Understand that when $$a$$ is negative, then the graph of the function is reflected over the $$x$$-axis.
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Anchor Problems

25-30 minutes

Problem 1

Consider the two functions below. 

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

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

  1. Create a table of values for each function and use it to graph each function in the coordinate plane. 
  2. Looking at the tables and the graphs, what changed from the parent function, $${f(x)}$$, to the new function, $${g(x)}$$? What is the same? 
  3. Look at the corresponding work in the Desmos activity Introduction to Transformations of Functions, slides 2–5. 

Guiding Questions

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References

Desmos Introduction to Transformations of Functions — Slides 2-5

Introduction to Transformations of Functions by is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 10, 2018, 4:22 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Predict what the graph would look like for each function below. Then refer to slide 10 in the Desmos activity Introduction to Transformations of Functions to try out other values for $$a$$.

a.   $${y={1\over2} |x|}$$

b.   $${y=-|x|}$$

c.   $${y=-10|x|}$$

d.   $$y=a|x|$$, for $$|a|>1$$

e.   $$y=a|x|$$, for $$|a|<1$$

Guiding Questions

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References

Desmos Introduction to Transformations of Functions — Slides 10-17

Introduction to Transformations of Functions by is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 10, 2018, 4:22 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

Sketch a graph of function $${ j(x)=-{1\over3} |x+4|-2}$$.

Guiding Questions

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

References

Desmos Introduction to Transformations of Functions — Slide 18

Introduction to Transformations of Functions by is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 10, 2018, 4:22 p.m..

Modified by Fishtank Learning, Inc.

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

The figure shows the graph of a function $$f$$ whose domain is the interval $${-2≤x≤2}$$.

  1. In (i)–(iii), sketch the graph of the given function and compare with the graph of $$f$$. Explain what you see. 

i.   $$g(x)=f(x)+2$$

ii.   $$h(x)=-f(x)$$

iii.   $$p(x)=f(x+2)$$

  1. The points labeled $$Q$$, $$O$$, and $$P$$ on the graph of $$f$$ have coordinates:

$$Q=(-2,-0.509)$$,     $$O=(0,-0.4)$$,    $$ P=(2,1.309)$$

What are the coordinates of the points corresponding to $$Q$$, $$O$$, and $$P$$ on the graphs of $$g$$, $$h$$, and $$p$$?

References

Illustrative Mathematics Transforming the Graph of a Function

Transforming the Graph of a Function, accessed on Nov. 19, 2017, 10:07 a.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.

Next

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

Lesson 15
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Lesson Map

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

Topic A: Piecewise Functions

Topic B: Absolute Value Functions

Topic C: Function Transformations

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