Curriculum  /  Math  /  9th Grade  /  Unit 5: Functions and Transformations  /  Lesson 12

Functions and Transformations

Lesson 12

Math

Unit 5

9th Grade

Lesson 12 of 16

Objective

Identify and describe vertical translations of functions.

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.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.
  • 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. Understand that the graphs of functions can undergo transformations that move the graph around the coordinate plane. 
  2. Identify and graph the parent functions of $${f(x)=x}$$, $${f(x)=|x|}$$, $${f(x)=x}^2$$, $${f(x)=x}^3$$, $${f(x)=\sqrt{x}}$$, and $${f(x)=\sqrt[3]{x}}$$.
  3. Identify when the graph of a function has been shifted vertically, both in a graph and in an equation.
  4. Describe how to shift a function’s graph vertically in a table of values or graphically.
  5. Draw graphs of functions that have been translated vertically. 
  6. Write equations, in function form, to represent graphs of functions that have been translated vertically using $$k$$ to represent the vertical translation (i.e., $$f(x)+k$$).

Tips for Teachers

Lessons 12–15 follow a similar format as they introduce the different transformations that can be applied to the graphs of functions in the coordinate plane. Desmos activities are featured in these lessons in order to capture the movements inherent in these transformations. 

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Anchor Problems

25-30 minutes

Problem 1

The functions below are parent functions. Graph each one. Use either technology or a table of values where needed. 

$${y=x}$$
$${y=x}^2$$
$${y=x}^3$$
$${y=|x|}$$
$${y=\sqrt{x}}$$
$${y=\sqrt[3]{x}}$$

Guiding Questions

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Problem 2

Consider the two functions below. 

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

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

  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 3–6. 

Guiding Questions

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References

Desmos Introduction to Transformations of Functions — Slides 3-6

Introduction to Transformations of Functions by Suzanne von Oy is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 10, 2018, 12:49 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

Predict what the tables and graph would look like for each function below. Then refer to slide 7 in the Desmos activity, Introduction to Transformations of Functions, to try out other values for $$k$$.

  1.   $${y=|x|+5}$$
  2.   $${y=|x|-1}$$
  3.   $$y=|x|+k$$

A different parent function and transformations are shown below. Predict what each graph would look like. Then refer to slides 8–14 in the Desmos activity. 

  1.   $${y=x^2}$$
  2.   $${y=x^2}+5$$
  3.   $${y=x^2}-1$$
  4.   $${y=x^2}+k$$

Guiding Questions

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References

Desmos Introduction to Transformations of Functions — Slides 7-14

Introduction to Transformations of Functions by Suzanne von Oy is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 10, 2018, 12:49 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

Given each parent function, $${f(x)}$$, below, sketch the transformed graph of the function $${g(x)}$$.

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.

  • EngageNY Mathematics Algebra I > Module 3 > Topic C > Lesson 17 — (Note, this lesson also covers vertical scaling and reflections; use only problems that involve vertical translations)
  • Desmos Absolute Value Translations — Slides 1-9
  • Continuous Everywhere But Differentiable Nowhere Function Transformations — (There are several documents with problems that can be used throughout the next few lessons. Functions Transformations 1 Basic Introduction; only use vertical translations)

Next

Identify and describe horizontal translations of functions.

Lesson 13
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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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