Write an exponential function to represent a relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.
Clarifications
Clarification 1: Within the Algebra 1 course, exponential functions are limited to the forms
, where
b is a whole number greater than 1 or a unit fraction, or
, where
.
Clarification 2: Within the Algebra 1 course, tables are limited to having successive nonnegative integer inputs so that the function may be determined by finding ratios between successive outputs.
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In middle grades, students solved problems involving percentages, including percent increases
and decreases and write equations that represent proportional relationships. In Algebra I, students
write exponential functions that model relationships characterized by having a constant percent
of change per unit interval. In later courses, students will further develop their understanding of
this feature of exponential functions.
- Provide opportunities to reference MA.912.AR.1.1 as students identify and interpret parts
of an exponential equation or expression as growth or decay and connect them to key
features of the graph.
- Problems include cases where the initial value is not given.
- Instruction includes guidance on how to determine the initial value or the percent rate of
change of an exponential function when it is not provided.
- For example, if the initial value of (0,3) is given, students can now write the
function as () = 3 . Guide students to choose a point on the curve that has
integer coordinates such as (2, 12). Lead them to substitute the point into their
function to find . Students should recognize that exponential functions are
restricted to positive values of , leading to the function () = 3(2) .
- Instruction includes interpreting percentages of growth/decay from exponential functions
expressed in the form () = and see that can be used to determine a percentage.
- For example, the function () = 500(0.72)x represents 16% growth of an
initial value.
- Guide students to discuss the meaning of the number 1.16 as a percent.
They should understand it represents 116%. Taking 116% of an initial
value increases the magnitude of that value. (Students can test this in a
calculator to confirm.) Taking this percentage repetitively leads to
exponential growth.
- For example, the function () = 500(0.72) represents 28% decay of an initial
value.
- Guide students to discuss the meaning of the number 0.72 as a percent.
They should understand it represents 72%. Taking 72% of an initial value
decreases the magnitude of that value. (Students can test this in a
calculator to confirm.) Taking this percentage repetitively leads to
exponential decay.
- For example, the function () = 500(1) represents an initial value that neither
grows nor decays as increases.
- Guide students to discuss the meaning of the number 1 when it comes to
growth/decay factors. They should understand it represents 100%. Taking
100% of an initial value causes the value to remain the same. (Students
can test this in a calculator to confirm.) Taking this percentage repetitively
leads to no change in the initial value (explaining the horizontal line that
shows when = 1 on the graph).
Common Misconceptions or Errors
- Students may not understand that exponential function values will eventually get larger
than those of any other polynomial functions because they do not fully understand the
impact of exponents on a value.
Strategies to Support Tiered Instruction
- Teacher provides students with a graphic displaying key terms within an exponential
function.
- Instruction includes comparing quadratic (polynomial) to exponential functions using
graphs or tables with whole-number inputs to show how exponential functions will
quickly exceed quadratic (polynomial) functions.
Instructional Tasks
Instructional Task 1 (MTR.4.1, MTR.5.1, MTR.7.1) - Karl and Simone were working on separate biology experiments. Each student documented
their cell population counts over time in the chart below.
- Part A. Do the number of cells in Simone’s experiment increase at a constant percentage
rate of change? If so, what is the percentage rate? If not, describe what is
happening to the number of cells. Does this change represent growth or decay?
Justify your answer.
- Part B. Write exponential functions to represent the relationship between the quantities
for each student’s experiment. In which experiment are the number of cells
changing more rapidly? Justify your answer.
- Part C. Graph these functions and determine their key features.
Instructional Task 2 (MTR.2.1, MTR.7.1) - The population of J-Town in 2019 was estimated to be 76,500 people with an annual rate of
increase of 2.4%.
- Part A. Write an equation to model future growth.
- Part B. What is the growth factor for J-Town?
- Part C. Use the equation to estimate the population in 2072 to the nearest hundred people.
Instructional Items
Instructional Item 1 - Write an exponential function that represents the graph below.
Instructional Item 2
- A forester has determined that the number of fir trees in a forest is decreasing by 3% per
year. In 2010, there were 13,000 fir trees in the forest. Write an equation that represents the
number of fir trees, , in terms of , the number of years since 2010.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Course Number1111 |
Course Title222 |
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7912070: | Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current)) |
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1200385: | Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current)) |
7912075: | Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current)) |
7912095: | Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current)) |
1207350: | Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current)) |
1200710: | Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current)) |
Related Access Points
Access Point Number |
Access Point Title |
MA.912.AR.5.AP.4 | Select an exponential function to represent two quantities from a graph or a table of values. |
Related Resources
Formative Assessments
Lesson Plans
Name |
Description |
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. |
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. |
Original Student Tutorial
Name |
Description |
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. |
Problem-Solving Tasks
Name |
Description |
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. |
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. |
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. |
Student Resources
Original Student Tutorial
Name |
Description |
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. |
Problem-Solving Tasks
Name |
Description |
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. |
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. |
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. |
Parent Resources
Problem-Solving Tasks
Name |
Description |
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. |
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. |
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. |