# MA.912.AR.5.7

Solve and graph mathematical and real-world problems that are modeled with exponential functions. Interpret key features and determine constraints in terms of the context.

### Examples

The graph of the function  can be transformed into the straight line y=5t+2 by taking the natural logarithm of the function’s outputs.

### Clarifications

Clarification 1: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; constant percent rate of change; end behavior and asymptotes.

Clarification 2: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation.

Clarification 3: Instruction includes understanding that when the logarithm of the dependent variable is taken and graphed, the exponential function will be transformed into a linear function.

Clarification 4: Within the Mathematics for Data and Financial Literacy course, problem types focus on money and business.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Domain
• Exponent
• Exponential function
• Function
• Set-builder notation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In Algebra I, students worked with exponential functions in limited forms. In Math for Data and Financial Literacy, students solve problems modeled with exponential functions.
• Instruction focuses on real-world contexts that require students to create a function as a tool to determine requested information or should provide the group a function that models the context.
• For example, students can use exponential decay when analyzing the historical depreciation of a car. In MA.912.AR.2.5, students find the straight-line depreciation. In this benchmark, students could find the percentage the car depreciates instead of the dollar amount (exponential depreciation) using the equation $y$ = A(1 − $r$)$x$, where A is the starting value of the car, $r$ is the percent of depreciation, $x$ is the elapsed time in years, and $y$ is the value of the car after $x$ years.
• Instruction includes making the connection to logarithms when working with investments formulas involving exponents.
• For example, when using the formula A = P(1 + $\frac{\text{r}}{\text{n}}$)$n$$t$ where $r$ is the interest rate expressed as a decimal, $n$ is the number of times the interest is compounded annually, and $t$ is the term of the account, students may need to use logarithms to find the value of a variable.
• For example, when using the formula A = P$e$$r$$t$, students will need to use the natural logarithm to isolate the variables $r$ and $t$.

• Instruction includes the use of technology and various representations of functions when solving problems. Given the complexity of the formulas in this course, it may be helpful for students to solve certain problems graphically rather than algebraically. Students can explore entering equations using graphing software.
• For example, if Sue deposited \$5,000 into an account that compounds interest monthly with an annual rate (APR) of 1.5%, students can determine how long it will take until the account reaches \$30,000 algebraically or graphically. To determine this graphically, students can graph the function $y$ = 5000(1 + $\frac{\text{0.015}}{\text{12}}$)12$t$ using technology. Students should realize that by finding where the graph reaches the $y$-value of 30,000, the corresponding $t$-value will provide how long it would take.
• For example, students can graph the equation 20000 =  to explore the relationship between the withdrawal amount and the time it takes to fully deplete a \$20,000 balance from an account that compounds quarterly at 2.1%.

• Problem types include using instances when periodic deposits are made into an account, and using formulas to find the term of systematic savings account. The formula to calculate the future value of a periodic deposit investment is B = , and the formula for the present value of a periodic deposit investment is P = .
• Problem types include using formulas to find the term of systematic withdrawals, which is when withdrawals are made at regular intervals. The formula to figure out the present value is P = .
• Be sure to highlight the relative ease and usefulness of graphing financial formulas to explore relationships. Graphs provide the visual data from an infinite number of calculations, saving students time and hassle when exploring relationships between variables.
• Instruction includes making connections to various forms of exponential functions to show their equivalency. Students should understand and interpret when solving the exponential in one form might be more useful than solving in another form depending on context.
• Instruction includes the use of $x$-$y$ notation and function notation.
• Instruction includes representing domain, range and intervals where the function is increasing, decreasing, positive or negative, using words, inequality notation, set-builder notation and interval notation.
• Words
If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number."
• Inequality notation
If the domain is all values of $x$ greater than 2, it can be represented as $x$ > 2.
• Set-builder notation
If the range is all values of $y$ less than or equal to zero, it can be represented as {$y$|$y$ ≤ 0} and is read as “all values of $y$ such that $y$ is less than or equal to zero.”
• Interval notation
If the domain is all values of $x$ less than or equal to 3, it can be represented as (−∞, 3]. If the domain is all values of $x$ greater than 3, it can be represented as (3, ∞). If the range is all values greater than or equal to −1 but less than 5, it can be represented as [−1, 5).

### Common Misconceptions or Errors

• Ensure students are plugging the correct information into the formulas and using technology correctly. Grouping symbols on graphic software can be tricky. Model the entry of a few formulas with grouping symbols for students before releasing them to explore their own.
• Students may confuse finding the logarithm and natural logarithm.

• Luisa deposited \$15,000 into a savings account, which compounds interest monthly with an annual rate (APR) of 1.15%. Each month, she withdraws \$350 from this account to pay for travel expenses.
• Part A. How long will it take until the account is fully depleted?
• Part B. How can she change the amount she withdrawals so her account lasts twice as long?

• Han wants to set up a periodic investment account to save for a future car purchase. His goal is to save \$40,000. The best account he’s found pays a 2.3% annual interest rate, compounded monthly. Use the periodic investment formula below to graph a relationship between the amount of his monthly periodic investment, P, and the time it will take to achieve his goal, $t$.

### Instructional Items

Instructional Item 1
• Mindy purchased a 3-year-old car for \$22,300. When the car was new, it sold for \$35,500. Find the depreciation rate to the nearest hundredth of a percent.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1202340: Precalculus Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1200384: Mathematics for Data and Financial Literacy (Specifically in versions: 2022 and beyond (current))
7912120: Access Mathematics for Data and Financial Literacy (Specifically in versions: 2022 - 2023, 2023 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))
1209315: Mathematics for ACT and SAT (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
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.

## Related Resources

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

## Formative Assessment

Interpreting Exponential Functions:

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

Type: Formative Assessment

## Lesson Plans

You’re Pulling My Leg – 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

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

## 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

## 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

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 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.

## MFAS Formative Assessments

Interpreting Exponential Functions:

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

## Original Student Tutorials Mathematics - Grades 9-12

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.

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.

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.

## Student Resources

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

## 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

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 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.

## Parent Resources

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

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 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.