Lesson 16 | Exponents and Exponential Functions | 9th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Interpreting Functions

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.

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

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.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.

Linear, Quadratic, and Exponential Models

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.

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.

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.

Foundational Standards

The foundational standards covered in this lesson

8.F.A.2

Functions

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.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.

Criteria for Success

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

Understand that linear functions have a constant rate of change and exponential functions have an increasing or decreasing rate of change.
Identify whether functions are linear or exponential in graphs, tables, equations, and situations.
Compare linear and exponential functions, identifying where the rates of change of exponential functions are greater than or less than linear functions.
Understand that an increasing exponential function will eventually exceed an increasing linear function.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson focuses on understanding and making connections across situations, graphs, and equations of linear and exponential functions. Students are not asked to generate equations, graphs, or contexts; they will work on those skills and concepts in the lessons that follow.

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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

Complete "Avi and Benita's Repair Shop" by Desmos.

Guiding Questions

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References

Desmos Avi and Benita's Repair Shop

Problem 2

Two equations are shown below.

A: $${y=0.01(2^x)}$$

B: $${y=100x}$$

Which equation represents Avi’s payment rule and which equation represents Benita’s payment rule?
Explain what each part of each equation means in context of the situation.
Describe, using features from the equations, how each function grows over time.

Guiding Questions

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Problem 3

Two equations and their graphs are shown.

Equation 1: $${y=5x}$$

Equation 2: $${y=0.5(2^x)}$$

a) Label each graph with the appropriate equation.

b) Describe the change in $$y$$ in each function as $$x$$ increases by $$1$$.

c) Describe the behavior of the exponential graph over time, as compared to the linear graph.

d) Over approximately what interval is the linear function greater than the exponential function?

Guiding Questions

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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

Use the function shown below to answer the questions that follow.

a. Which equation matches the graph?

i. $${y=3(2^x )}$$

ii. $${y=2(3^x )}$$

iii. $${y=4x+2}$$

iv. $${y=2x+4}$$

b. Explain why you chose that equation in part (a).

c. What is the rate of change of this function?

d. Suppose the graph of $${y=150x}$$ was added to the graph above. Which function is greater over the domain interval $${1<x<5}$$? Will this function always be greater over all values of the domain? Explain your reasoning.

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.

Include examples where students are given different representations of linear and exponential functions (tables, graphs, equations), and must match them together

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 5 — Opening Exercise and Example 2
Illustrative Mathematics Equal Factors over Equal Intervals
Illustrative Mathematics Boiling Water
Illustrative Mathematics Equal Differences over Equal Intervals 2
Mathematics Vision Project: Secondary Mathematics Two Module 2: Linear & Exponential Functions — 2.2 "Ready", 2.3 "Linear, Exponential, or Neither?", "Ready"

Lesson 15

Lesson 17

Lesson Map

Topic A: Exponent Rules, Expressions, and Radicals

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

8.EE.A.1

Add and subtract polynomial expressions using properties of operations.

A.APR.A.1

Multiply polynomials using properties of exponents and properties of operations.

A.APR.A.1

Solve mathematical applications of exponential expressions.

8.EE.A.1

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.

N.RN.B.3

Simplify radical expressions.

N.RN.A.2

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.

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

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.

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

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.

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

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

Exponents and Exponential Functions

Objective

Common Core Standards

Core Standards

Interpreting Functions

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

Seeing Structure in Expressions

Foundational Standards

Functions

Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Additional Practice

Lesson Map

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