The given exponential function is:

$y = 500(1.3)^{-3x}$

Step 1: Identify Growth or Decay

• The general form of an exponential function is:

  $y = a(b)^x$

  where:

  • $a$ is the initial value.
  • $b$ is the base (growth or decay factor).
  • If $b > 1$, it represents exponential growth.
  • If $0 < b < 1$, it represents exponential decay.

• In the given function, $b = 1.3$, which is greater than 1. However, the exponent is negative ($-3x$), which effectively changes the base to:

  $(1.3)^{-1} = \frac{1}{1.3} \approx 0.7692$

  Since the effective base is less than 1, the function represents exponential decay.

Step 2: Determine the Percentage Rate of Decrease

• The decay factor per unit of $x$ is:

  $b^{|k|} = (1.3)^3$

  Calculating $(1.3)^3$:

  $1.3^3 = 2.197$

  The decay factor per unit $x$ is:

  $\frac{1}{2.197} \approx 0.455$

  The percentage rate of decrease is:

  $(1 - 0.455) \times 100 = 54.5\%$

Thus, the function represents exponential decay with a 54.5% decrease per unit of $x$.

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Related Questions:

1. How does a negative exponent affect an exponential function?
2. What is the general formula to find the percentage rate of exponential decay?
3. If the exponent were positive ($y = 500(1.3)^{3x}$), what would be the growth rate?
4. How do you convert exponential equations into logarithmic form?
5. Can an exponential function ever cross the x-axis? Why or why not?

Tip:

Always check the exponent sign in an exponential function; a negative exponent flips growth into decay and vice versa.