For the function f, for each increase in the value of a by c, where c is a positive constant, the value of f(a) increasesby a factor of 27. Which of the following equivalent forms of the function f displays - as a coefficient of r?

To solve this problem, let's break it down step by step.

### Problem Breakdown:
1. The function \( f(a) \) increases by a factor of 27 each time \( a \) increases by a constant \( c \).
2. This implies that the function is likely an exponential function of the form:
   \[
   f(a) = r \cdot b^{\frac{a}{c}}
   \]
   where \( r \) is a coefficient and \( b \) is the base.

### Analysis:
- Given that each time \( a \) increases by \( c \), the function \( f(a) \) is multiplied by 27:
  \[
  f(a + c) = 27 \cdot f(a)
  \]

  Substituting the exponential form:
  \[
  r \cdot b^{\frac{a+c}{c}} = 27 \cdot r \cdot b^{\frac{a}{c}}
  \]

  Simplify the exponent on the left:
  \[
  r \cdot b^{\frac{a}{c} + 1} = 27 \cdot r \cdot b^{\frac{a}{c}}
  \]

  Since \( r \) and \( b^{\frac{a}{c}} \) are common factors on both sides, we can cancel them out:
  \[
  b = 27
  \]

  Therefore, the function can be written as:
  \[
  f(a) = r \cdot 27^{\frac{a}{c}}
  \]

### Conclusion:
The equivalent form of the function \( f(a) \) that shows \( r \) as a coefficient is:
\[
f(a) = r \cdot 27^{\frac{a}{c}}
\]

This form clearly displays \( r \) as the coefficient of \( 27^{\frac{a}{c}} \).

### Follow-Up
Do you want more details or have any questions about the solution? 

Here are some related questions for further exploration:

1. What is the impact on the function if the factor of increase was different from 27?
2. How would the function change if \( f(a) \) decreased by a factor of 27 instead of increasing?
3. What is the derivative of this function with respect to \( a \)?
4. How does the base 27 relate to the logarithmic form of the function?
5. What happens to \( f(a) \) as \( a \) approaches infinity?

**Tip:** Exponential functions often model scenarios where quantities grow or shrink by a consistent percentage or factor over time, such as compound interest or population growth.

Explore how an exponential function changes when a constant factor of 27 is applied to its growth with detailed steps and examples. Suitable for high school students studying exponential functions and growth patterns.

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