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Chapter 6: Problem 2

Use the equation \(y=47(1-0.12)^{x}\) to answer each question. a. Does this equation model an increasing or decreasing pattern? (Th b. What is the rate of increase or decrease? c. What is the \(y\)-value when \(x\) is 13 ? d. What happens to the \(y\)-values as the \(x\)-values get very large?

Short Answer

Expert verified
a. Decreasing pattern. b. 12% decrease. c. \(y \approx 11.0843\). d. \(y\) approaches 0 as \(x\) increases.

Step by step solution

01

Identify Pattern Type

In the given equation, we have the term \((1-0.12)^x\). Since the number inside the parenthesis is less than 1 (i.e., 0.88), the equation models a decreasing pattern. Exponential functions of the form \(a(b)^x\) where \(0 < b < 1\) result in a decreasing pattern.
02

Determine Rate of Change

The rate of decrease is given by the number being subtracted from 1 inside the exponent, which is 0.12 in this case. Therefore, the rate of decrease is 12%.
03

Calculate y-value for x = 13

To find the \(y\)-value when \(x = 13\), substitute 13 into the equation: \[ y = 47(1-0.12)^{13} = 47(0.88)^{13} \] Calculate \((0.88)^{13}\), and then multiply by 47 to get the \(y\) value.Using a calculator, \((0.88)^{13} \approx 0.2359\). Therefore, \[ y = 47 \times 0.2359 \approx 11.0843 \].
04

Predict Behavior as x Increases

As \(x\) becomes very large, \((0.88)^x\) approaches 0 because 0.88 raised to the power of a large number becomes very small. Therefore, the \(y\)-values approach 0 as \(x\) increases indefinitely.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decreasing Pattern
When examining exponential functions, one of the key aspects to determine is whether the function models a decreasing or increasing pattern. This is based on the base value of the exponent. For the function given, \(y=47(1-0.12)^{x}\), the term \((1-0.12)^x\) simplifies to \(0.88^x\). Because the base, 0.88, is less than 1, each increase in \(x\) results in a smaller value of \(0.88^x\). Therefore, the function exhibits a decreasing pattern. In simpler terms, this means that as you plug in larger numbers for \(x\), the output \(y\) becomes smaller. This predictable decrease characterizes exponential decay when the base is between 0 and 1.
Rate of Change
The rate of change in an exponential function, especially for an exponential decay model, is crucial because it tells us how quickly values diminish over time. In our function, \(y=47(1-0.12)^{x}\), the rate of change is dictated by the value subtracted from 1 inside the parentheses, which is 0.12. This subtraction indicates a 12% decrease each time \(x\) increments by 1. To conceptualize this:
  • Every time \(x\) increases by 1, \(y\) decreases by 12%.
  • This consistent fractional reduction results in the smooth, exponential decline of the function.
Understanding the rate of change is pivotal for predicting future values and grasping how swiftly the function's quantities diminish.
Exponential Decay
Exponential decay is a specific type of change where values decrease at a consistent percentage rate over equal increments of time or another variable. The given function, \(y=47(1-0.12)^x\), is a perfect example of exponential decay. Here's why it's crucial:
  • Exponential decay means that the quantity reduces rapidly at first, then slows down as it approaches zero.
  • This behavior is contrasted with linear decrease, where values reduce steadily rather than exponentially.
  • In practical terms, exponential decay models phenomena like radioactive decay, population decrease in limited environments, and depreciation of assets.
Comprehending exponential decay helps us foresee long-term behavior of systems that diminish based on a fixed rate.
Asymptotic Behavior
Asymptotic behavior in mathematics refers to how a function behaves as the input grows very large or very small. For our function, \(y=47(0.88)^x\), as \(x\) becomes extremely large, the term \(0.88^x\) approaches zero. Here's why this matters:
  • It implies that \(y\) will get infinitesimally close to zero but never actually reach it.
  • This is a hallmark of exponential decay functions where the values continue shrinking but at a slower rate as time progresses.
  • This leads to a horizontal asymptote at \(y = 0\), indicating the boundary that the function will not cross.
Visualizing the asymptotic behavior allows better understanding of limits and future function behavior, especially crucial in fields like calculus and real-world modelling where long-term predictions matter.

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