 # Comparing Linear and Exponential Models

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Lesson Plans and Worksheets for Algebra I
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Common Core For Algebra I

Examples, solutions, and videos to help Algebra I students learn how to create models and understand the differences between linear and exponential models that are represented in different ways.

### New York State Common Core Math Module 3, Algebra I, Lessons 21

Worksheets for Algebra I, Module 3, Lesson 21 (pdf)

Lesson 21 Summary

Suppose that the input-output pairs of a bivariate dataset have the following property: for every two inputs that are a given difference apart, the difference in their corresponding outputs is constant. Then an appropriate model for that dataset could be a linear function.

Suppose that the input-output pairs of a bivariate dataset have the following property: for every two inputs that are a given difference apart, the quotient of their corresponding outputs is constant. Then an appropriate model for that dataset could be an exponential function.

An increasing exponential function will eventually exceed any linear function. That is, if f(x) = abx is an exponential function with a > 0 and b > 1, and g(x) = mx + k is any linear function, then there is a real number M such that for all x > M, then f(x) > g(x). Sometimes this is not apparent in a graph displayed on a graphing calculator; that is because the graphing window does not show enough of the graph to show the sharp rise of the exponential function in contrast with the linear function.

Problem Set Sample Solutions

For each table in Problems 1–6, classify the data as describing a linear relationship, an exponential growth relationship, an exponential decay relationship, or neither. If the relationship is linear, calculate the constant rate of change (slope), and write a formula for the linear function that models the data. If the function is exponential, calculate the common quotient for input values that are distance 1 apart, and write the formula for the exponential function that models the data. For each linear or exponential function found, graph the equation y = f(x).

Exit Ticket
Here is a classic riddle: Mr. Smith has an apple orchard. He hires his daughter, Lucy, to pick apples and offers her two payment options.
Option A: per bushel of apples picked.
Option B: cent for picking one bushel, cents for picking two bushels, cents for picking three bushels, and so on, with the amount of money tripling for each additional bushel picked.
a. Write a function to model each option.
b. If Lucy picks six bushels of apples, which option should she choose?
c. If Lucy picks bushels of apples, which option should she choose?
d. How many bushels of apples does Lucy need to pick to make option B better for her than option A?

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