To determine the exponential function of the form [tex]\( f(x) = a(b)^x \)[/tex] that models this situation, we need to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Given points:
- The initial length when [tex]\( x = 0 \)[/tex] is 16 inches, so the point is [tex]\( (0, 16) \)[/tex].
- After one copy, when [tex]\( x = 1 \)[/tex], the length is 12 inches, so the point is [tex]\( (1, 12) \)[/tex].
Step-by-step solution:
1. Determine [tex]\( a \)[/tex]:
The function [tex]\( f(x) = a(b)^x \)[/tex] passes through the point [tex]\( (0, 16) \)[/tex].
When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = a(b)^0 = a \cdot 1 = a
\][/tex]
Since [tex]\( f(0) = 16 \)[/tex], we have:
[tex]\[
a = 16
\][/tex]
2. Determine [tex]\( b \)[/tex]:
The function also passes through the point [tex]\( (1, 12) \)[/tex].
When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = a(b)^1 = a \cdot b
\][/tex]
Given [tex]\( f(1) = 12 \)[/tex], and we already know [tex]\( a = 16 \)[/tex], we can substitute these values into the equation:
[tex]\[
12 = 16 \cdot b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = \frac{12}{16} = 0.75
\][/tex]
Putting it all together, we have the exponential function:
[tex]\[
f(x) = 16 \cdot (0.75)^x
\][/tex]
Therefore, the correct values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
a = 16, \quad b = 0.75
\][/tex]
Your answer should be:
[tex]\[
f(x) = 16 \cdot (0.75)^x
\][/tex]
