Let's analyze the given exponential growth function [tex]\( f(x) = 81(1.024)^x \)[/tex]:
1. Initial Value of the Function:
In an exponential function of the form [tex]\( f(x) = a \cdot b^x \)[/tex], the initial value is represented by the coefficient [tex]\( a \)[/tex]. This coefficient is the value of the function when [tex]\( x = 0 \)[/tex].
Here, the coefficient [tex]\( a \)[/tex] is 81. Therefore, the initial value of the function is
[tex]\[ \boxed{81} \][/tex]
2. Growth Factor:
The growth factor in an exponential function [tex]\( f(x) = a \cdot b^x \)[/tex] is given by the base [tex]\( b \)[/tex] of the exponent. This base indicates how much the function is multiplied by for each increase by 1 in [tex]\( x \)[/tex].
In the given function, the base [tex]\( b \)[/tex] is 1.024. Therefore, the growth factor of the function is
[tex]\[ \boxed{1.024} \][/tex]
3. Growth Rate (as a Percent):
The growth rate as a percent can be derived from the growth factor by subtracting 1 from the growth factor and then multiplying by 100.
Here, the growth factor is 1.024, so the growth rate as a percent is computed as:
[tex]\[ (1.024 - 1) \times 100 = 0.024 \times 100 = 2.4 \][/tex]
Therefore, the growth rate of the function as a percent is
[tex]\[ \boxed{2.4} \% \][/tex]
So, summarizing the answers:
- The initial value of the function is [tex]\( \boxed{81} \)[/tex].
- The growth factor is [tex]\( \boxed{1.024} \)[/tex].
- The growth rate as a percent is [tex]\( \boxed{2.4} \% \)[/tex].
