Here’s the solution for the given function $f(x) = 3(0.78)^x$:

1. Identify the factor. Is it growth or decay?

• The factor is 0.78.
• Since $0.78 < 1$, the function represents exponential decay.

2. Identify the initial value.

• The initial value is the coefficient of the exponential factor, which is 3.
• This represents the value of the function when $x = 0$, i.e., $f(0) = 3$.

3. Find the rate.

• The decay factor $0.78$ relates to the rate $r$ of decay using the formula:
  $\text{Decay Factor} = 1 - r$
• Here, $0.78 = 1 - r$. Solving for $r$:
  $r = 1 - 0.78 = 0.22$
• Therefore, the rate of decay is 22%.

Summary of Answers:

1. Factor: $0.78$ → Exponential decay.
2. Initial Value: $3$.
3. Rate of Decay: $22\%$.

Let me know if you need further clarification or detailed steps for any part!

Related Questions:

1. How do you rewrite this function in terms of the rate $r$?
2. How do you determine whether a function is growth or decay from a graph?
3. If $f(x) = 3(0.78)^x$, what is $f(5)$?
4. How does changing the initial value affect the graph of the function?
5. How long does it take for the function's value to decrease to half its initial value?

Tip: In exponential functions, the factor determines whether the function grows ($>1$) or decays ($<1$). Always compare it to $1$.