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Learn how to model, analyze, and predict data using exponential and logarithmic functions. Identify patterns and relationships between quantities. Practice fitting exponential functions to datasets.
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Curve Fitting with Exponential and Logarithmic Models Essential Questions • How do we model data by using exponential and logarithmic functions? • How do we use exponential and logarithmic models to analyze and predict? Holt McDougal Algebra 2 Holt Algebra 2
Analyzing data values can identify a pattern, or repeated relationship, between two quantities. Look at this table of values for the exponential function f(x) = 2(3x).
Remember! For linear functions (first degree), first differences are constant. For quadratic functions, second differences are constant, and so on. Notice that the ratio of each y-value and the previous one is constant. Each value is three times the one before it, so the ratio of function values is constant for equally spaced x-values. This data can be fit by an exponential function of the form f(x) = abx.
First Second Differences Differences 81 24 54 36 3 = = = = 24 54 2 36 16 Identifying Exponential Data Determine whether f is an exponential function of x of the form f(x) = abx. If so, find the constant ratio. 2. 1. 8 12 18 27 1 2 3 4 1 1 1 4 6 9 Second differences are constant; f is a quadratic function of x. Ratio This data set is exponential, with a constant ratio of 1.5.
First Second Differences Differences 13.5 4 9 3 6 = = = = 9 2 4 2.6 6 Identifying Exponential Data Determine whether f is an exponential function of x of the form f(x) = abx. If so, find the constant ratio. 4. 3. 5 5 5 5 1.34 2 3 4.5 0.66 1 1.5 Ratio First differences are constant; y is a linear function of x. This data set is exponential, with a constant ratio of 1.5.
