To determine which graph represents the sequence defined by the function [tex]\( f(x) = 3 \cdot 2^{x-1} \)[/tex], let's compute some values of the function for a few points. We will evaluate the function at [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], [tex]\( x = 3 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( x = 5 \)[/tex] to understand how the sequence progresses.
1. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \cdot 2^{1-1} = 3 \cdot 2^0 = 3 \cdot 1 = 3
\][/tex]
So, the point corresponding to [tex]\( x = 1 \)[/tex] is [tex]\( (1, 3) \)[/tex].
2. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 3 \cdot 2^{2-1} = 3 \cdot 2^1 = 3 \cdot 2 = 6
\][/tex]
So, the point corresponding to [tex]\( x = 2 \)[/tex] is [tex]\( (2, 6) \)[/tex].
3. Calculate [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = 3 \cdot 2^{3-1} = 3 \cdot 2^2 = 3 \cdot 4 = 12
\][/tex]
So, the point corresponding to [tex]\( x = 3 \)[/tex] is [tex]\( (3, 12) \)[/tex].
4. Calculate [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = 3 \cdot 2^{4-1} = 3 \cdot 2^3 = 3 \cdot 8 = 24
\][/tex]
So, the point corresponding to [tex]\( x = 4 \)[/tex] is [tex]\( (4, 24) \)[/tex].
5. Calculate [tex]\( f(5) \)[/tex]:
[tex]\[
f(5) = 3 \cdot 2^{5-1} = 3 \cdot 2^4 = 3 \cdot 16 = 48
\][/tex]
So, the point corresponding to [tex]\( x = 5 \)[/tex] is [tex]\( (5, 48) \)[/tex].
Summarizing these results, we have the following sequence of points for [tex]\( x \)[/tex] ranging from [tex]\( 1 \)[/tex] to [tex]\( 5 \)[/tex]:
[tex]\[
\{(1, 3), (2, 6), (3, 12), (4, 24), (5, 48)\}
\][/tex]
The graph of the sequence will plot these points on the coordinate plane. Based on the computed points, we see an exponential pattern, which means the graph will show an exponential growth as [tex]\( x \)[/tex] increases.
