Comparing Linear and Exponential Models
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, Lesson 21
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Lessons for Algebra I
Common Core For Algebra I
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).
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