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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.
Standard #: MA.912.AR.5.4
Standard Information
Standard Clarifications
Clarification 1: Within the Algebra 1 course, exponential functions are limited to the forms begin mathsize 12px style f open parentheses x close parentheses equals a b to the power of x end style, where b is a whole number greater than 1 or a unit fraction, or begin mathsize 12px style f open parentheses x close parentheses equals a open parentheses 1 plus-or-minus r close parentheses to the power of x end style, where begin mathsize 12px style 0 less than r less than 1 end style.

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.

General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Algebraic Reasoning
Standard: Write, solve and graph exponential and logarithmic equations and functions in one and two variables.
Date Adopted or Revised: 08/20
Status: State Board Approved
Standard Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Exponential

 

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 f(x) = 3bx  . 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 b. Students should recognize that exponential functions are restricted to positive values of b, leading to the function f(x) = 3(2)x .
  • Instruction includes interpreting percentages of growth/decay from exponential functions expressed in the form f(x) = abxand see that b can be used to determine a percentage. 
    • For example, the function f(x) = 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 f(x) = 500(0.72)x  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 f(x) = 500(1)x represents an initial value that neither grows nor decays as x 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 b = 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.
    Table
    • 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. 
    Graph


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, N, in terms of t, 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
Related Access Points
  • 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
  • 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
  • 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
  • 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.
  • 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.
  • Basketball Rebounds # 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.
MFAS Formative Assessments
Original Student Tutorials Mathematics - Grades 9-12
  • 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.
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