# Standard 5: Write, solve and graph exponential and logarithmic equations and functions in one and two variables.

General Information
Number: MA.912.AR.5
Title: Write, solve and graph exponential and logarithmic equations and functions in one and two variables.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.912.AR.5.AP.2
Solve one-variable equations involving logarithms or exponential expressions. Identify any extraneous solutions.
MA.912.AR.5.AP.3
Given a real-world context, identify an exponential function as representing growth or decay.
MA.912.AR.5.AP.4
Select an exponential function to represent two quantities from a graph or a table of values.
MA.912.AR.5.AP.5
Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents.
MA.912.AR.5.AP.6
Given a table, equation or written description of an exponential function, select the graph that represents the function.
MA.912.AR.5.AP.7
Given a mathematical and/or real-world problem that is modeled with exponential functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model.
MA.912.AR.5.AP.8
Given an equation of a logarithmic function, select the graph of that function.
MA.912.AR.5.AP.9
Given a mathematical and/or real-world problem that is modeled with logarithmic functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

Interpreting Exponential Functions:

Students are asked to interpret parameters of an exponential function in context.

Type: Formative Assessment

Trees in Trouble:

Students are asked to write a function that represents an annual loss of 3 percent each year.

Type: Formative Assessment

Exponential Growth:

Students are given two functions, one represented verbally and the other by a table, and are asked to compare the rate of change in each in the context of the problem.

Type: Formative Assessment

Graphing an Exponential Function:

Students are asked to graph an exponential function and to determine if the function is an example of exponential growth or decay, describe any intercepts, and describe the end behavior of the graph.

Type: Formative Assessment

Students are asked to write and solve an equation that models an exponential relationship between two variables.

Type: Formative Assessment

Comparing Functions - Exponential:

Students are asked to use technology to graph exponential functions and then to describe the effect on the graph of changing the parameters of the function.

Type: Formative Assessment

Loss of Fir Trees:

Students are asked to sketch a graph that depicts the exponential decline in the population of fir trees in a forest.

Type: Formative Assessment

Exponential Functions - 2:

Students are asked to identify the percent rate of change of a given exponential function.

Type: Formative Assessment

Exponential Functions - 1:

Students are asked to identify the percent rate of change of a given exponential function.

Type: Formative Assessment

Case In Point:

Students are asked to explain the relationship between the set of solutions and the graph of an exponential equation.

Type: Formative Assessment

What Is the Function Rule?:

Students are asked to write function rules for sequences given tables of values.

Type: Formative Assessment

Writing an Exponential Function From a Table:

Students are asked to write an exponential function represented by a table of values.

Type: Formative Assessment

Writing an Exponential Function From a Description:

Students are asked to write an exponential function from a written description of an exponential relationship.

Type: Formative Assessment

Writing an Exponential Function From Its Graph:

Students are asked to write an exponential function given its graph.

Type: Formative Assessment

## Lesson Plans

Youâ€™re Pulling My Leg â€“ or Candy!:

Students will explore how exponential growth and decay equations can model real-world problems. Students will also discover how manipulating the variables in an exponential equation changes the graph. Students will watch a Perspectives Video to see how exponential growth is modeled in the real world.

Type: Lesson Plan

My Geometry Classroom:

Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

Type: Lesson Plan

The Copernicus' Travel:

This lesson uses Inverse Trigonometric Ratios to find acute angle measures in right triangles. Students will analyze the given information and determine the best method to use when solving right triangles. The choices reviewed are Trigonometric Ratios, The Pythagorean Theorem, and Special Right Triangles.

Type: Lesson Plan

Which Function?:

This activity has students apply their knowledge to distinguish between numerical data that can be modeled in linear or exponential forms. Students will create mathematical models (graph, equation) that represent the data and compare these models in terms of the information they show and their limitations. Students will use the models to compute additional information to predict future outcomes and make conjectures based on these predictions.

Type: Lesson Plan

Exponential Graphing Using Technology:

This lesson is teacher/student directed for discovering and translating exponential functions using a graphing app. The lesson focuses on the translations from a parent graph and how changing the coefficient, base and exponent values relate to the transformation.

Type: Lesson Plan

## Original Student Tutorials

Exponential Functions Part 3: Decay:

Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 2: Growth:

Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 1:

Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.

Type: Original Student Tutorial

Creating Exponential Functions:

Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Type: Perspectives Video: Professional/Enthusiast

Making Candy: Illuminating Exponential Growth:

<p>No need to sugar coat it: making candy involves math and muscles. Learn how light refraction and exponential growth help make candy colors just right!</p>

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Kites, Wind, and Speed:

Lofty ideas about kites helped power a kayak from California to Hawaii.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

Solving Quadratic Equation Using Loh's Method:

Unlock an effective teaching strategy for solving quadratic equations in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

What functions do two graph points determine?:

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Two Points Determine an Exponential Function II:

This problem solving tasks asks students to find the values of points on a graph.

Two Points Determine an Exponential Function I:

This problem solving task asks students to graph a function and find the values of points on a graph.

Snail Invasion:

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.

Rumors:

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Do two points always determine an exponential function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Carbon 14 Dating in Practice I:

In the task "Carbon 14 Dating" the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died, and as this task shows, that is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.

Carbon 14 Dating:

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

This task asks students to analyze data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. This variant of the task is not scaffolded; for a more scaffolded version, see Basketball Bounces, Assessment Variation 1.

Students are asked to select the best model for a given context and use the model to make predictions. This task assesses students’ modeling skills. Students are tasked to distinguish between situations that can be modeled with linear and exponential functions and recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

## Tutorial

Graphing Exponential Equations:

This tutorial will help you to learn about exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Exponential Functions Part 3: Decay:

Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 2: Growth:

Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 1:

Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.

Type: Original Student Tutorial

Creating Exponential Functions:

Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Making Candy: Illuminating Exponential Growth:

<p>No need to sugar coat it: making candy involves math and muscles. Learn how light refraction and exponential growth help make candy colors just right!</p>

Type: Perspectives Video: Professional/Enthusiast

Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

What functions do two graph points determine?:

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Two Points Determine an Exponential Function II:

This problem solving tasks asks students to find the values of points on a graph.

Two Points Determine an Exponential Function I:

This problem solving task asks students to graph a function and find the values of points on a graph.

Snail Invasion:

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.

Rumors:

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Do two points always determine an exponential function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Carbon 14 Dating in Practice I:

In the task "Carbon 14 Dating" the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died, and as this task shows, that is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.

Carbon 14 Dating:

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

## Tutorial

Graphing Exponential Equations:

This tutorial will help you to learn about exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

## Perspectives Video: Expert

Problem Solving with Project Constraints:

<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Making Candy: Illuminating Exponential Growth:

<p>No need to sugar coat it: making candy involves math and muscles. Learn how light refraction and exponential growth help make candy colors just right!</p>

Type: Perspectives Video: Professional/Enthusiast

Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

What functions do two graph points determine?:

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Two Points Determine an Exponential Function II:

This problem solving tasks asks students to find the values of points on a graph.

Two Points Determine an Exponential Function I:

This problem solving task asks students to graph a function and find the values of points on a graph.

Snail Invasion:

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.

Rumors:

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Do two points always determine an exponential function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Carbon 14 Dating in Practice I:

In the task "Carbon 14 Dating" the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died, and as this task shows, that is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.

Carbon 14 Dating:

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.