Lesson 14 | Functions and Transformations | 9th Grade Mathematics

Objective

Identify and describe vertical scaling of functions, including reflections over the $$x$$-axis.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Building Functions

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.

Interpreting Functions

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

The foundational standards covered in this lesson

8.F.B.4

Functions

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.G.A.3

Geometry

8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

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

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Consider the two functions below.

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

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

Create a table of values for each function and use it to graph each function in the coordinate plane.
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?
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

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

Modified by Fishtank Learning, Inc.

Problem 3

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

Guiding Questions

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References

Desmos Introduction to Transformations of Functions — Slide 18

Modified by Fishtank Learning, Inc.

Problem Set

A set of suggested resources or problem types that teachers can turn into a 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

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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

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. $$p(x)=f(x+2)$$

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.

EngageNY Mathematics Algebra I > Module 3 > Topic C > Lesson 17 — Problem Set (use the problems not used in Lesson 12)
Illustrative Mathematics Building a quadratic function from f(x)=x^2
Continuous Everywhere But Differentiable Nowhere Function Transformations — Functions Transformations 3 Vertical Stretching

Lesson 13

Lesson 15

Lesson Map

Topic A: Piecewise Functions

Write equations and graph linear relationships with different domains.

A.CED.A.1 A.CED.A.2 A.CED.A.3 F.IF.B.5 F.IF.C.7

Graph piecewise functions presented algebraically and write piecewise functions from graphs.

F.IF.A.2 F.IF.B.4 F.IF.C.7.B

Evaluate a piecewise function written algebraically, with and without context.

A.SSE.A.1.A F.IF.A.1 F.IF.A.2

Write and graph piecewise functions from contextual situations.

A.CED.A.2 A.CED.A.3 F.IF.B.5

Graph, evaluate, and write step functions, with and without context.

A.CED.A.3 F.IF.A.1 F.IF.C.7.B

Topic B: Absolute Value Functions

Graph absolute value functions in the coordinate plane and identify domain and range.

F.BF.B.3 F.IF.B.4 F.IF.B.5 F.IF.C.7.B F.IF.C.9

Identify the solutions to an absolute value equation.

A.REI.A.1 A.REI.D.11

Solve absolute value equations algebraically and identify extraneous solutions.

A.REI.B.3

Identify the solution(s) to an absolute value inequality.

A.REI.A.1 A.REI.B.3 A.REI.D.11 F.IF.B.4 F.IF.C.7.B

Solve absolute value inequalities algebraically and verify graphically.

A.REI.B.3

Identify solutions to a system of absolute value and linear functions graphically and algebraically.

A.REI.A.1 A.REI.C.6 A.REI.D.11

Topic C: Function Transformations

Identify and describe vertical translations of functions.

F.BF.B.3 F.IF.B.4 F.IF.B.5

Identify and describe horizontal translations of functions.

F.BF.B.3 F.IF.B.5

Identify and describe vertical scaling of functions, including reflections over the $$x$$-axis.

F.BF.B.3 F.IF.B.5

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

F.BF.B.3 F.IF.B.5

Apply transformations to functions.

F.BF.B.3

Functions and Transformations

Objective

Common Core Standards

Core Standards

Building Functions

Interpreting Functions

Foundational Standards

Functions

Geometry

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Problem Set

Target Task

References

Additional Practice

Lesson Map

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