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
| Properties of exponents | Polynomial, binomial, trinomial | Degree of term or polynomial | Leading coefficient |
| Exponential expression | Radical | Root | Radicand |
| Index | Recursive formula | Explicit formula | Sequence |
| Sequence notation, $${a_n}$$ | Fibonacci sequence | Arithmetic sequence | Geometric sequence |
| Common difference | Common ratio | Exponential growth/decay function | Compound interest |
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<sup>-5</sup> = 3<sup>-3</sup> = 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<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)³ = 5(<sup>1/3</sup>)³ to hold, so (5<sup>1/3</sup>)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)<sup>t</sup>, y = (0.97)<sup>t</sup>, y = (1.01 <sup>12t</sup>, y = (1.2)<sup>t/10</sup>, 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 x<sup>4</sup> — y<sup>4</sup> 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<sup>ct</sup> = 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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Homework 3.5: Applications of Exponentials Name: _____ Math 3 1. Find a bank account balance if the account starts with $100, has an annual rate of 4%, and the money left in the account for 12 years. 2. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by
Applications of Exponential Functions. Get a hint. Periodic Compound Interest. Click the card to flip 👆. A = P (1 + r/n)^nt. Click the card to flip 👆. 1 / 17.
2. Likewise, the exponential function ex is one of the most important functions used in mathematics, statistics, and many fields of science. For a variety of reasons, the base e turns out to be the most natural base to use for an exponential function. Consequently, the function is known as the natural exponential function.
Show Details. Use tools learned previously to solve real-world problems. MAT.ALG.906.0708 (Applications of Exponential Functions - Algebra) MAT.ALG.906.0708 (Applications of Exponential Functions - Calculus)
5.8: Applications of Exponential and Logarithmic Functions
HOMEWORK ON EXPONENTIAL FUNCTIONS _ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the function value. If the result is irrational, round your answer to the nearest thousandth. ... $2613 invested for 3 years at 5% compounded quarterly A) $3030.28 B) $3024.87 C) $3034.93 D) $3033.05 26) 9.
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.
By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, \(g(x)=x^3\) does not represent an exponential function because the base is an independent variable. In fact, \(g(x)=x^3\) is a power function.
Microsoft Word - 3.5 transformations of exponential functions worksheet.docx Author: Trevor Jensen Created Date: 20141016234017Z ...
Homework 3.5: Applications of Exponentials Math 3 Name: Find a bank account balance if the account starts with $100, has an annual rate of 4%, and the money left in the account for 12 years. 2. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985.
e0.1171t 3.2342 30.92e0.1171t 100 D 100. 30.92e0.1171t D D 100 t. D t 3 D 30.92e0.1171t, 3 ≤t 7 258 Chapter 3 Exponential and Logarithmic Functions Example 1 This section shows students real-world applications of logarithmic and exponential functions. t D Year (3 ↔ 2003) Digital Television Households (in millions) 34567 20 40 60 80 100 ...
Study with Quizlet and memorize flashcards containing terms like Which of the following represents an exponential function?, f(1/2) = 6(9)^1/2, and more.
Application of Exponential and Logarithmic Growth functions
As we mentioned in Section 6.1, exponential and logarithmic functions are used to model a wide variety of behaviors in the real world. In the examples that follow, note that while the applications are drawn from many different disciplines, the mathematics remains essentially the same. Due to the applied nature of the problems we will examine in ...
In this activity students will practice their knowledge with exponential growth and decay functions. Skills required:Identify difference between growth and decay in word problems.Set up exponential function to model exponential growth and decay: f(t)=a(1+r)^tUse knowledge of exponential functions and logarithms to solve for an exponent (6 problems)For each of the 12 problems, students will ...
Divide both sides by the coefficient, 80, to isolate the exponential expression. 2500 80 = 80 80e0.12t 31.25 = e0.12t. Rewrite the equation in logarithmic form. 0.12t = ln(31.25) Divide both sides by 0.04 to isolate t; then use your calculator and its natural log function to evaluate the expression and solve for t.
Exponential & logarithmic functions | Algebra (all content)
A(t) = A0e−kt (3.5.1) where A(t) is the amount of material left at time t,A0 is the amount present at t = 0, and k is a constant that can be determined based on the half-life of the material. 0.5A0 = A0e−kth (3.5.2) where th is the half-life. To solve this equation for k, we would first divide on both sides by A0: