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Answer :
To determine whether the equation represents exponential growth, exponential decay, or neither, and to find the starting value of the function, let's analyze the given function:
The function is given by:
[tex]\[ f(t) = 220(1.06)^t \][/tex]
This is a standard form of an exponential function, where:
- The coefficient [tex]\( 220 \)[/tex] is the starting value or the initial amount.
- The base [tex]\( 1.06 \)[/tex] is the growth or decay factor.
### Step 1: Identify the Starting Value
The starting value is the amount when [tex]\( t = 0 \)[/tex]. In this function:
[tex]\[ f(t) = 220(1.06)^t \][/tex]
When [tex]\( t = 0 \)[/tex]:
[tex]\[ f(0) = 220(1.06)^0 = 220 \times 1 = 220 \][/tex]
Thus, the starting value is [tex]\( 220 \)[/tex].
### Step 2: Determine if the Function Represents Growth or Decay
In an exponential function [tex]\( a(b)^t \)[/tex]:
- If the base [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
- If [tex]\( b = 1 \)[/tex], the function represents neither growth nor decay.
In our equation:
[tex]\[ b = 1.06 \][/tex]
Since [tex]\( 1.06 > 1 \)[/tex], this means the function represents exponential growth.
### Conclusion
- The equation represents exponential growth.
- The starting value of the function is [tex]\( 220 \)[/tex].
The function is given by:
[tex]\[ f(t) = 220(1.06)^t \][/tex]
This is a standard form of an exponential function, where:
- The coefficient [tex]\( 220 \)[/tex] is the starting value or the initial amount.
- The base [tex]\( 1.06 \)[/tex] is the growth or decay factor.
### Step 1: Identify the Starting Value
The starting value is the amount when [tex]\( t = 0 \)[/tex]. In this function:
[tex]\[ f(t) = 220(1.06)^t \][/tex]
When [tex]\( t = 0 \)[/tex]:
[tex]\[ f(0) = 220(1.06)^0 = 220 \times 1 = 220 \][/tex]
Thus, the starting value is [tex]\( 220 \)[/tex].
### Step 2: Determine if the Function Represents Growth or Decay
In an exponential function [tex]\( a(b)^t \)[/tex]:
- If the base [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
- If [tex]\( b = 1 \)[/tex], the function represents neither growth nor decay.
In our equation:
[tex]\[ b = 1.06 \][/tex]
Since [tex]\( 1.06 > 1 \)[/tex], this means the function represents exponential growth.
### Conclusion
- The equation represents exponential growth.
- The starting value of the function is [tex]\( 220 \)[/tex].
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