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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.
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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.
