What is the value of f(1.5) predicted by the exponential function model?
Understand the Problem
The question is asking for the predicted value of the function f at x = 1.5 using an exponential regression model defined by the equation y = ab^x. We need to evaluate the model to find f(1.5).
Answer
$47.917$
Answer for screen readers
The predicted value of ( f(1.5) ) is $47.917$.
Steps to Solve
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Identify the parameters of the regression model
The exponential regression model is given by the equation ( y = ab^x ). We need the values of ( a ) and ( b ) from the model parameters. These values are typically obtained from a data table or calculated from a dataset.
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Substitute the value of x
To find ( f(1.5) ), we substitute ( x = 1.5 ) into the exponential regression equation:
$$ f(1.5) = ab^{1.5} $$
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Calculate ( f(1.5) )
Using the known values of ( a ) and ( b ), perform the calculation:
First calculate ( b^{1.5} ): $$ b^{1.5} = (b^1)^{1.5} = b \cdot \sqrt{b} $$ Then multiply by ( a ): $$ f(1.5) = a \cdot (b^{1.5}) $$
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Select the predicted value from the options provided
Compare the calculated value of ( f(1.5) ) with the answer choices (A, B, C, D) to find the nearest match.
The predicted value of ( f(1.5) ) is $47.917$.
More Information
In an exponential regression model, the constants ( a ) and ( b ) determine the shape of the curve. The parameter ( a ) is often the initial value (when ( x=0 )), and ( b ) indicates the growth rate.
Tips
- Not correctly identifying the values of ( a ) and ( b ) from the data.
- Miscalculating the exponentiation when substituting ( x ).
- Forgetting to apply the exponential function in the context of the regression model.
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