Math Problem Statement
Solution
The given equation is . Let us break it down to identify the requested components:
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Original amount (): This is the initial value of the function when . In this case, .
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Growth factor (): This is the base of the exponential term. Here, .
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Rate of increase (): The growth factor is related to the rate of increase using the formula: Solving for : r = b - 1 = 5 - 1 = 4 \quad \text{(or 400% as a percentage)}.
Final Answer:
- Original amount ():
- Growth factor ():
- Rate of increase (): (or ).
Would you like me to explain these steps further or provide examples?
Relative Questions:
- How do you determine the growth factor in exponential functions?
- What is the difference between the growth factor and the rate of increase?
- How do exponential growth and decay differ in terms of their formulas?
- Can the original amount () ever be negative? What does that imply?
- How would the function change if the growth factor was less than 1?
Tip:
Remember that in exponential functions, indicates growth, while indicates decay!
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Growth and Decay
Formulas
Exponential Function: y = a · b^x
Growth Factor Relation: b = 1 + r
Rate of Increase: r = b - 1
Theorems
Properties of Exponential Functions
Suitable Grade Level
Grades 8-10