Question
Given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease. \[ y=8600(0.11)^{x} \] Answer Attempt sout of 2 Growth ~ \% increase Submit Answer
Ask by Salazar Franklin. in the United States
Feb 25,2025
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Answer
The function y = 8600(0.11)^x represents exponential decay with an 89% decrease per unit increase in x.
Solution
The function is given by y = 8600(0.11)^x. Notice that the base of the exponential term is 0.11, which is less than 1. This tells us that as x increases, y gets multiplied by 0.11 each time, meaning the function represents exponential decay.
To find the percentage rate of decay:
• Subtract the base from 1: 1 − 0.11 = 0.89.
• Convert this to a percentage: 0.89 × 100% = 89%.
Thus, the function represents exponential decay with an 89% decrease per unit increase in x.
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Simplify this solution Extra Insights
This exponential function represents decay since the base, 0.11, is less than 1. In exponential decay, the value of \( y \) decreases as \( x \) increases. To determine the percentage rate of decrease, subtract the base from 1 and multiply by 100: \[ (1 - 0.11) \times 100 = 89\% \] Thus, the function indicates an 89% decrease. When looking at exponential functions more broadly, remember that growth functions will always have a base greater than 1 (like 1.1, which indicates a 10% growth). Conversely, bases between 0 and 1 will show decay, as in this case where you see the percentage drop. Recognizing this pattern can help dissect exponential behaviors in various scenarios!
