To determine which graph represents the function [tex]\( f(x) = 2^{x-1} + 2 \)[/tex], we need to analyze its characteristics and behavior.
### Step 1: Understanding the Function
The function [tex]\( f(x) = 2^{x-1} + 2 \)[/tex] can be broken down into parts:
- Base Function: [tex]\( 2^{x} \)[/tex] is the standard exponential function with base 2.
- Shifting: [tex]\( 2^{x-1} \)[/tex] indicates a horizontal shift by 1 unit to the right because of the [tex]\( x-1 \)[/tex] term.
- Vertical Shift: Adding 2 to the result means the entire graph is shifted up by 2 units.
### Step 2: Determining Key Points
Let's determine some key points on the graph:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2^{0-1} + 2 = 2^{-1} + 2 = \frac{1}{2} + 2 = 2.5
\][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2^{1-1} + 2 = 2^{0} + 2 = 1 + 2 = 3
\][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^{2-1} + 2 = 2^{1} + 2 = 2 + 2 = 4
\][/tex]
### Step 3: Behavior of the Function
- Asymptote: The function [tex]\( f(x) \)[/tex] will approach [tex]\( y = 2 \)[/tex] as [tex]\( x \)[/tex] approaches negative infinity, due to the horizontal and vertical shift.
- Growth: Since the base of the exponential part is 2, the function will grow exponentially as [tex]\( x \)[/tex] increases.
### Step 4: Plotting Points and Analyzing Graph
Now that we have key points and understand the behavior:
- At [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] is approximately 2.5.
- At [tex]\( x = 1 \)[/tex], [tex]\( y \)[/tex] is 3.
- At [tex]\( x = 2 \)[/tex], [tex]\( y \)[/tex] is 4.
- There is a horizontal asymptote at [tex]\( y = 2 \)[/tex].
### Conclusion
Based on the key points and the asymptotic behavior, we need to select the graph that:
- Grows exponentially.
- Has a horizontal asymptote at [tex]\( y = 2 \)[/tex].
- Shows the specified key points.
Observing the graphs provided (A or B) and identifying the one that fits this precise behavior will help us choose the correct representation of the function [tex]\( f(x) = 2^{x-1} + 2 \)[/tex].
If you describe the graphs and their behavior, I can help you determine which one accurately represents the function.
