Lesson 20 | Exponents and Exponential Functions | 9th Grade Mathematics

Objective

Write, graph, and evaluate exponential decay functions.

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.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Interpreting Functions

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.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.

Interpreting Functions

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

Linear, Quadratic, and Exponential Models

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.

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.

Criteria for Success

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

Understand that an exponential decay function has a decreasing rate of change, represented by a value between $$0$$ and $$1$$, compared to an exponential growth function that has an increasing rate of change.
Identify exponential decay functions in equations, tables, and graphs.
Analyze an exponential decay function in equation form to identify features in the graph of the function including the $$y$$-intercept, domain, range, rate of change, etc.
Analyze the graph of an exponential decay function to write an equation for the function.

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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

Compare and contrast functions $${{f(x)}}$$ and $${{g(x)}}$$ by answering the questions below.

$${{f(x)}}=\left({1\over2}\right)^x$$

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

a. What happens to the values of $${{f(x)}}$$ and $${{g(x)}}$$ as $$x$$ increases?

b. What are the domains of $${{f(x)}}$$ and $${{g(x)}}$$?

c. What are the ranges of $${{f(x)}}$$ and $${{g(x)}}$$?

d. Create a table of values for each function and use them to sketch a graph of each function.

Guiding Questions

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Problem 2

Consider the exponential graph below.

Create a table of values for the function.
Does the function represent exponential growth or exponential decay? Explain your reasoning.
What transformations were applied to the graph of this function? Explain your reasoning.
Determine the equation for this function.

Guiding Questions

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Problem 3

Sketch a graph of each function shown below.

$${f(x)=2(3)^{-x}}$$

$${g(x)=\left(1\over2\right)^x-3}$$

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

Problem 1

Graph the exponential function $${f(x)=-\left({1\over2}\right)^x}$$.

Problem 2

Write the equation for the graph shown below.

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 problems where students graph exponential decay functions, including those with transformations

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 7 — Exercise 1
Kuta Software Free Algebra 2 Worksheets Graphing Exponential Functions
Desmos Card Sort: Exponentials — (This can be used on the computer or adapted to be done with paper materials)
Desmos Polygraph: Exponentials — (This can be used as is, or adapted for other uses to fit the lesson)

Lesson 19

Lesson 21

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

Building Functions

Interpreting Functions

Interpreting Functions

Linear, Quadratic, and Exponential Models

Foundational Standards

Functions

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Problem 1

Problem 2

Additional Practice

Lesson Map

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