ALEX Classroom Resource

  

Comparing Quadratic and Exponential Functions

  Classroom Resource Information  

Title:

 Comparing Quadratic and Exponential Functions

URL:

 https://aptv.pbslearningmedia.org/resource/im20-math-ep4-64/comparing-quadratic-and-exponential-functions/

Content Source:

 PBS
Type: Audio/Video

Overview:

In this video lesson, students investigate how quantities that grow quadratically compare to those that grow exponentially. They discover the reason that increasing exponential functions also eventually surpass increasing quadratic functions. By examining successive quotients for each type of function, students see that the outputs of quadratic functions are not multiplied by the same factor each time the input increases by one. In fact, these successive quotients get smaller as the inputs increase, while the outputs of the exponential function have the same multiplier. As they compare the two types of functions, they develop their understanding of quadratic expressions and how the shape of the graph differs between the two types of functions.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
  • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
  • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

  • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
Teacher Vocabulary:
  • Linear functions
  • Exponential functions
  • Constant rate of change
  • Constant percent rate of change
  • Intervals
  • Percentage of growth
  • Percentage of decay
  • Slope-intercept form of a line
Knowledge:
Students know:
  • Key components of linear and exponential functions.
  • Properties of operations and equality
Skills:
Students are able to:
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).
Understanding:
Students understand that:
  • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
  • Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
28. Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. [Algebra I with Probability, 26]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a quantity increasing exponentially and a quantity increasing linearly or quadratically.
  • Construct graphs and tables that demonstrate the exponential function will exceed the linear or quadratic function at some point.
  • Present a convincing argument that this must be true for all polynomial functions.
Teacher Vocabulary:
  • Increasing exponentially
  • Increasing quadratically
  • Increasing linearly
Knowledge:
Students know:
  • Techniques to graph and create tables for exponential, linear, and quadratic functions.
Skills:
Students are able to:
  • Accurately create graphs and tables for exponential, linear, and quadratic functions.
  • Use the graphs and tables to present a convincing argument that the exponential function eventually exceeds the linear and quadratic function.
Understanding:
Students understand that:
  • Exponential functions grow at a faster rate than linear and quadratic functions after some point in their domain.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
  • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
  • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

  • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
Teacher Vocabulary:
  • Linear functions
  • Exponential functions
  • Constant rate of change
  • Constant percent rate of change
  • Intervals
  • Percentage of growth
  • Percentage of decay
Knowledge:
Students know:
  • Key components of linear and exponential functions.
  • Properties of operations and equality
Skills:
Students are able to:
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).
Understanding:
Students understand that:
  • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
  • Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.24.1: Define linear function and exponential function.
ALGI.24.2: Distinguish between graphs of a line and an exponential function.
ALGI.24.3: Identify the graph of an exponential function.
ALGI.24.4: Identify the graph of a line.
ALGI.24.5: Plot points on a coordinate plane from a given table of values. a.
ALGI.24.6: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.7: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.8: Apply rules for adding, subtracting, multiplying, and dividing integers. b.
ALGI.24.9: Define constant rate of change as slope.
ALGI.24.10: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.11: Recognize the calculated difference is the constant rate of change.
ALGI.24.12: Apply rules for adding, subtracting, multiplying, and dividing integers. c.
ALGI.24.13: Define exponential growth and decay.
ALGI.24.14: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.15: Apply the rules of multiplication and division of integers.

Prior Knowledge Skills:
  • Recognize ordered pairs.
  • Identify ordered pairs.
  • Recognize linear equations.
  • Recall how to solve problems using the distributive property.
  • Define linear and nonlinear functions, slope, and y-intercept.
  • Analyze the graph to determine the rate of change.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
26. Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a quantity increasing exponentially and a quantity increasing as a polynomial function (e.g., linearly, quadratically),
  • Construct graphs and tables that demonstrate the exponential function will exceed the polynomial function at some point.
  • Present a convincing argument that this must be true for all polynomial functions.
Teacher Vocabulary:
  • Increasing exponentially
  • Increasing linearly
  • Polynomial functions
Knowledge:
Students know:
  • Techniques to graph and create tables for exponential and polynomial functions.
Skills:
Students are able to:
  • Accurately create graphs and tables for exponential and polynomial functions.
  • Use the graphs and tables to present a convincing argument that the exponential function eventually exceeds the polynomial function.
Understanding:
Students understand that:
  • Exponential functions grow at a faster rate than polynomial functions after some point in their domain.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.26.1: Define a polynomial function and parabola.
ALGI.26.4: Compare the y-values by looking at the same x-value in a variety of tables or graphs.
ALGI.26.3: Identify the graph of an exponential function.
ALGI.26.4: Identify the graph of a line.
ALGI.26.5: Plot points on a coordinate plane from a given table of values.
ALGI.26.6: Identify the graph of a quadratic function.

Prior Knowledge Skills:
  • Create a graph to model a real-word situation.
  • Compare and contrast the relationship between two quantities in a graph.
  • Compare and contrast the differences between linear and nonlinear functions.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.
Tags: exponential, graphs of functions, quadratic, successive quotients
License Type: Public Domain
For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses
AccessibilityVideo resources: includes closed captioning or subtitles
Comments

This resource contains student practice problems and student task statements.

  This resource provided by:  
Author: Kristy Lacks
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