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All Things Algebra®

Algebra 1 Unit 6: Exponents & Exponential Functions

This unit includes 55 pages of guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the topics listed in the description below.

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• Additional Information
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This unit contains the following topics:

• Adding and Subtracting Monomials (Review of Combine Like Terms) • Multiplying Monomials (Product Rule) • Multiplying Monomials (Power Rule) • Dividing Monomials (Quotient Rule) • Negative Exponents • Review of all Exponent Rules • Scientific Notation • Graphing Exponential Functions • Exponential Growth and Decay • Geometric Sequences • Simplifying Radicals: Square Roots and Cube Roots • Simplifying Square Roots with Variables

This unit does not contain activities.

This is the guided notes, homework assignments, quizzes, study guide, and unit test only.  For suggested activities to go with this unit, check out the ATA Activity Alignment Guides .

This resource is included in the following bundle(s):

Algebra 1 Curriculum Algebra 1 Curriculum (with Activities)

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Math  /  9th Grade  /  Unit 6: Exponents and Exponential Functions

Exponents and Exponential Functions

Students extend their understanding of properties of exponents to include rational exponents and radicals, and investigate rates of change in linear and exponential sequences and functions.

Unit Summary

In Unit 6, students compare linear and exponential functions in novel ways to reveal new information about and applications of each one. They also extend their understanding of properties of exponents from 8th Grade Math to include rational exponents and radicals.  

In Topic A, students recall the properties of exponents and operations that enable complex-looking expressions to be written more simply. Students are briefly introduced to polynomials, and they add, subtract, and multiply polynomials using properties. Students learn how to write rational exponential expressions from radicals, and vice versa, and they hone their skills in simplifying and computing with both forms.

In Topic B, students are introduced to sequences, specifically arithmetic and geometric sequences. They write recursive and explicit formulas for both types of sequences, focusing on precision in their notation and language. By identifying the common difference or ratio and looking at tables and graphs of sequences, students make connections between the rate of growth of a sequence and whether it is linear or exponential. 

In Topic C, students make more explicit connections between increasing or decreasing rates of change and exponential functions. They graph exponential functions and transformations of exponential functions, and they write equations for those functions using key features from the graphs. Students look at several real-world applications of exponential growth and decay, in particular the concept of compound interest.

Pacing: 24 instructional days (22 lessons, 1 flex day, 1 assessment day)

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The following assessments accompany Unit 6.

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Post-Unit Assessment Answer Key

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

• Standards that each question aligns to
• Purpose of each question: spiral, foundational, mastery, developing
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit 
• Lesson(s) that assessment points to

Internalization of Trajectory of Unit

• Read and annotate “Unit Summary.”
• Notice the progression of concepts through the unit using “Unit at a Glance.”
• Essential understandings
• Connection to assessment questions

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

• Exponential expressions with rational exponents follow the same properties of exponents as integer exponents. In addition, a rational exponent, $${a\over b}$$ , defines a radical where $$a$$  is the exponent of the radicand and $$b$$  is the index of the radical. For example,  $${10^{5\over6}}$$ can be written as  $${\sqrt[6]{10^5}}$$ .
• An arithmetic sequence consists of terms that increase or decrease linearly by a constant value, called the common difference. Comparatively, a geometric sequence consists of terms that increase or decrease exponentially by a constant factor called the common ratio. Both arithmetic and geometric sequences can be represented by recursive and explicit formulas.
• Exponential growth and decay functions can be used to model situations that involve a constant percent rate of growth or decay over time, such as compound interest. An exponentially increasing quantity will always eventually exceed a linearly increasing quantity.

Terms and notation that students learn or use in the unit

The materials, representations, and tools teachers and students will need for this unit

• Technology for graphing
• Calculators

Topic A: Exponent Rules, Expressions, and Radicals

Use exponent rules to analyze and rewrite expressions with non-negative exponents.

Add and subtract polynomial expressions using properties of operations.

Multiply polynomials using properties of exponents and properties of operations. 

Solve mathematical applications of exponential expressions.

Use negative exponent rules to analyze and rewrite exponential expressions.

8.EE.A.1 A.SSE.A.2

Define rational exponents and convert between rational exponents and roots. 

N.RN.A.1 N.RN.A.2

Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.

Simplify radical expressions.

Multiply and divide rational exponent expressions and radical expressions.

N.RN.A.2 N.RN.B.3

Add and subtract rational exponent expressions and radical expressions.

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Topic B: Arithmetic and Geometric Sequences

Describe and analyze sequences given their recursive formulas.

F.BF.A.2 F.IF.A.2 F.IF.A.3

Write recursive formulas for sequences, including the Fibonacci sequence.

Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.

F.BF.A.2 F.LE.A.2

Write explicit rules for arithmetic sequences and translate between explicit and recursive formulas.

F.BF.A.2 F.IF.A.3 F.LE.A.2

Write explicit rules for geometric sequences and translate between explicit and recursive formulas.

Topic C: Exponential Growth and Decay

Compare rates of change in linear and exponential functions shown as equations, graphs, and situations.

A.SSE.A.1 F.IF.C.9 F.LE.A.1 F.LE.A.3

Write linear and exponential models for real-world and mathematical problems.

A.SSE.A.1 F.LE.A.1 F.LE.A.2 F.LE.B.5

Graph exponential growth functions and write exponential growth functions from graphs.

F.BF.B.3 F.IF.C.7.E

Write exponential growth functions to model financial applications, including compound interest.

F.IF.C.8.B F.LE.A.2 F.LE.B.5

Write, graph, and evaluate exponential decay functions.

F.BF.B.3 F.IF.C.7.E F.IF.C.8.B F.LE.A.1.C

 Identify features of exponential decay in real-world problems.

F.IF.C.8.B F.LE.A.1.C

Solve exponential growth and exponential decay application problems.

F.IF.C.8.B F.LE.A.1 F.LE.A.2

Common Core Standards

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Building Functions

F.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 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.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.

Expressions and Equations

8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3^-5 = 3^-3 = 1/3³ = 1/27.

High School — Number and Quantity

N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)³ = 5(^1/3)³ to hold, so (5^1/3)³ must equal 5.

N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

N.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Interpreting Functions

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.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 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.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01 ^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.

F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Linear, Quadratic, and Exponential Models

F.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

F.LE.A.1.C — Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

F.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context 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.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.

7.EE.A.1 — Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

8.EE.C.7 — Solve linear equations in one variable.

8.F.A.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

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.

Ratios and Proportional Relationships

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

The Number System

8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Future Standards

Standards in future grades or units that connect to the content in this unit

F.BF.B.4 — Find inverse functions.

F.BF.B.5 — Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

F.LE.A.4 — For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

A.SSE.B.4 — Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. 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.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

CCSS.MATH.PRACTICE.MP6 — Attend to precision.

CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Functions and Transformations

Quadratic Functions and Solutions

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Alg1.5 Introduction to Exponential Functions

In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare them to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts with quantities that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. They learn that the output of an increasing exponential function is eventually greater than the output of an increasing linear function for the same input.

Students view these new types of relationships as functions and employ the notation and terminology of functions (for example, dependent and independent variables). They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don’t represent a particular context, observing the effect of different values of \(a\) and \(b\) on the graph of the function \(f\) represented by \(f(x)=ab^x\) .  The context of credit (both in terms of loans and savings) is used through several lessons.

Looking at Growth

• 1 Growing and Growing
• 2 Patterns of Growth

A New Kind of Relationship

• 3 Representing Exponential Growth
• 4 Understanding Decay
• 5 Representing Exponential Decay
• 6 Analyzing Graphs
• 7 Using Negative Exponents

Exponential Functions

• 8 Exponential Situations as Functions
• 9 Interpreting Exponential Functions
• 10 Looking at Rates of Change
• 11 Modeling Exponential Behavior
• 12 Reasoning about Exponential Graphs (Part 1)
• 13 Reasoning about Exponential Graphs (Part 2)

Percent Growth and Decay

• 14 Recalling Percent Change
• 15 Functions Involving Percent Change
• 16 Compounding Interest
• 17 Different Compounding Intervals
• 18 Expressed in Different Ways

Comparing Linear and Exponential Functions

• 19 Which One Changes Faster?
• 20 Changes over Equal Intervals

Putting It All Together

• 21 Predicting Populations

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Unit 6 – Exponential Algebra and Functions

Exponential Increase and Decrease

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

Geometric Sequences

Equivalent Exponential Expressions

Simplifying Fractions Involving Exponents

Zero and Negative Exponents

More Work with Exponent Properties

Introduction to Exponential Functions

Percent Review

Percent Increase and Decrease

Exponential Models Based on Percent Growth

Constructing Exponential Functions

Linear Versus Exponential Functions

Unit Review

Unit 6 Review

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit 6 Assessment – Form A

EDITABLE ASSESSMENT

Unit 6 Assessment – Form B

Unit 6 Exit Tickets

Unit 6 Mid-Unit Quiz – Form A

U06.AO.01 – Investment Options Modeling Task

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Exponential Functions Unit Algebra 1 TEKS

$ 10.00

An 8-day Exponential Functions TEKS-Aligned complete unit including graphing exponential functions, writing exponential functions, exponential growth and decay, and applying exponential functions.

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An 8-day Exponential Functions (TEKS-Aligned) complete unit including graphing exponential functions, writing exponential functions, exponential growth and decay, and applying exponential functions.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills. You can reach your students and teach the standards without all of the prep and stress of creating materials!

Learning Focus:

• graph exponential functions and identify key features including domain and range
• interpret the meaning of the values a and b in functions of the form y = ab^x
• write exponential functions to describe growth and decay

Standards: TEKS: A.9A, A.9B, A.9C, A.9D AND A.9E; Looking for CCSS-Aligned Resources? Grab the CCSS-Aligned Exponential Functions Unit . Please don’t purchase both as there is overlapping content.

What is included in the Algebra 1 TEKS Exponential functions Unit?

• Unit Overviews
• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.
• Student Handouts
• Student-friendly guided notes are scaffolded to support student learning.
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.
• Independent Practice
• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice.
• Assessments
• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.
• Answer Keys
• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our Exponential Functions Activity Bundle for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

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• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

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This resource is often paired with:

Exponential Functions Activity Bundle Algebra 1

Alg1.5 Introduction to Exponential Functions

In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare them to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts with quantities that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. They learn that the output of an increasing exponential function is eventually greater than the output of an increasing linear function for the same input.

Students view these new types of relationships as functions and employ the notation and terminology of functions (for example, dependent and independent variables). They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don’t represent a particular context, observing the effect of different values of \(a\) and \(b\) on the graph of the function \(f\) represented by \(f(x)=ab^x\) .  The context of credit (both in terms of loans and savings) is used through several lessons.

Looking at Growth

• 1 Growing and Growing
• 2 Patterns of Growth

A New Kind of Relationship

• 3 Representing Exponential Growth
• 4 Understanding Decay
• 5 Representing Exponential Decay
• 6 Analyzing Graphs
• 7 Using Negative Exponents

Exponential Functions

• 8 Exponential Situations as Functions
• 9 Interpreting Exponential Functions
• 10 Looking at Rates of Change
• 11 Modeling Exponential Behavior
• 12 Reasoning about Exponential Graphs (Part 1)
• 13 Reasoning about Exponential Graphs (Part 2)

Percent Growth and Decay

• 14 Recalling Percent Change
• 15 Functions Involving Percent Change
• 16 Compounding Interest
• 17 Different Compounding Intervals
• 18 Expressed in Different Ways

Comparing Linear and Exponential Functions

• 19 Which One Changes Faster?
• 20 Changes over Equal Intervals

Putting It All Together

• 21 Predicting Populations