A scientist is attempting to model measured data of a growth process using a base function. The base function is given by \(f(x) = e^x\).
(a) Determine the new function when the data indicates the growth is scaled vertically by a factor of 2.
(b) Adjust the function if the measurements show a horizontal shift to the right by 3 units.
(c) Compare the residuals of both models with the measured data and discuss which transformation provides a better fit and why.
Question
Determine the new function when the data indicates the growth is scaled vertically by a factor of 2.
Hint
Hint
Hint
Multiply the function by the scaling factor.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The new function is \(2e^x\).
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Question
Adjust the function if the measurements show a horizontal shift to the right by 3 units.
Hint
Hint
Hint
Replace \(x\) with \(x-3\) in the function.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Considering the scaling from part (a), the complete transformation is \(2e^{(x-3)}\).
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Answer Explanation
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Question
Compare the residuals of both models with the measured data and discuss which transformation provides a better fit and why.
Hint
Hint
Hint
Discuss the closeness of the model predictions to the measured data based on residual differences.
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Complete Answer
The model \(2e^{(x-3)}\) is a better fit because the horizontal shift aligns the exponential growth phase with the data’s growth phase, reducing systematic errors in the residuals. For the model \(2e^x\), the residuals might show a pattern of consistently positive or negative values at the beginning or end of the data range, indicating a systematic deviation. By shifting the function horizontally, we aim to reduce this systematic pattern and achieve more randomly distributed residuals, which is a sign of a better fit.
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Answer Explanation
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