Pregunta
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Given the following exponential function, identify whether the change represents
growth or decay, and determine the percentage rate of increase or decrease.

Ask by Dunn Nichols. in the United States
Nov 12,2024

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Responder

To determine if an exponential function represents growth or decay, check if the base is greater than 1 for growth or between 0 and 1 for decay. The percentage rate is calculated as for growth and for decay.

Solución

To determine whether an exponential function represents growth or decay, and to find the percentage rate of increase or decrease, we typically look at the function in the form:
where:
  • is the initial value,
  • is the base of the exponential function,
  • is time.
  1. Identify Growth or Decay:
    • If , the function represents exponential growth.
    • If , the function represents exponential decay.
  2. Determine the Percentage Rate:
    • The percentage rate of increase or decrease can be calculated using the formula:
      • For growth:
      • For decay:
If you provide the specific exponential function you are working with, I can help you analyze it further!

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Extra Insights

Exponential functions can indicate either growth or decay based on the base of the exponent. If the base is greater than 1, the function represents growth; if it is between 0 and 1, it represents decay. To determine the percentage rate, you can use the formula , where is the base of the exponential function. For instance, if the function is , it shows growth, and the percentage rate of increase is . Conversely, for , it signifies decay with a percentage rate of decrease of .
If you need practical examples and visualizations, consider graphing these functions using software or graphing calculators—they offer fantastic insights into how the function behaves over time! You’ll see the sharp climb of growth functions and the graceful decline of decay functions, making it easier to grasp these concepts visually.

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