Answer :
To determine which graph represents the function [tex]\( f(x) = 2^{x-1} + 2 \)[/tex], let's analyze and describe the characteristics of this function step-by-step.
### Step 1: Understand the Form of the Function
The given function is [tex]\( f(x) = 2^{x-1} + 2 \)[/tex]. This is an exponential function of the form [tex]\( a^{x-b} + c \)[/tex], where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 2 \)[/tex]
### Step 2: Analysis of the Function [tex]\( 2^{x-1} \)[/tex]
The base function we start with is [tex]\( 2^x \)[/tex]. Shifting this function to the right by 1 unit gives us [tex]\( 2^{x-1} \)[/tex].
### Step 3: Vertical Translation
Adding 2 to the function [tex]\( 2^{x-1} \)[/tex] will shift the entire graph up by 2 units. Thus:
- The horizontal asymptote of the function [tex]\( 2^{x-1} \)[/tex] at [tex]\( y = 0 \)[/tex] will be shifted to [tex]\( y = 2 \)[/tex].
### Step 4: Key Characteristics and Behavior of the Function
1. Asymptote: The graph will have a horizontal asymptote at [tex]\( y = 2 \)[/tex]. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^{x-1} \)[/tex] approaches 0, so [tex]\( f(x) \)[/tex] approaches 2.
2. Y-intercept: When [tex]\( x=0 \)[/tex]:
[tex]\[ f(0) = 2^{0-1} + 2 = 2^{-1} + 2 = \frac{1}{2} + 2 = 2.5 \][/tex]
3. Growth: Since it’s an exponential function with base greater than 1, it will grow rapidly as [tex]\( x \)[/tex] increases.
### Step 5: Sketching the Graph
- Horizontal asymptote at [tex]\( y = 2 \)[/tex].
- Y-intercept at [tex]\( (0, 2.5) \)[/tex].
- For [tex]\( x > 1 \)[/tex], the function grows exponentially because [tex]\( 2^{x-1} \)[/tex] increases rapidly.
### Step 6: Plot a Few More Points for Verification
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2^{1-1} + 2 = 2^0 + 2 = 1 + 2 = 3 \][/tex]
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2^{-1-1} + 2 = 2^{-2} + 2 = \frac{1}{4} + 2 = 2.25 \][/tex]
### Summary of Important Points:
- Horizontal Asymptote: [tex]\( y = 2 \)[/tex].
- Y-intercept: [tex]\( (0, 2.5) \)[/tex].
- Point (1, 3).
- Point (-1, 2.25).
- Rapid Exponential Growth for [tex]\( x > 1 \)[/tex].
Based on these characteristics, you should now be able to identify the correct graph among the given options. You are looking for a graph that:
- Approaches but never touches [tex]\( y = 2 \)[/tex] for negative [tex]\( x \)[/tex] values.
- Passes through the point [tex]\( (0, 2.5) \)[/tex].
- Passes through the point [tex]\( (1, 3) \)[/tex].
- Exhibits rapid exponential growth as [tex]\( x \)[/tex] increases.
Look for these features in the provided graph options to select the correct graph for [tex]\( f(x) = 2^{x-1} + 2 \)[/tex].
### Step 1: Understand the Form of the Function
The given function is [tex]\( f(x) = 2^{x-1} + 2 \)[/tex]. This is an exponential function of the form [tex]\( a^{x-b} + c \)[/tex], where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 2 \)[/tex]
### Step 2: Analysis of the Function [tex]\( 2^{x-1} \)[/tex]
The base function we start with is [tex]\( 2^x \)[/tex]. Shifting this function to the right by 1 unit gives us [tex]\( 2^{x-1} \)[/tex].
### Step 3: Vertical Translation
Adding 2 to the function [tex]\( 2^{x-1} \)[/tex] will shift the entire graph up by 2 units. Thus:
- The horizontal asymptote of the function [tex]\( 2^{x-1} \)[/tex] at [tex]\( y = 0 \)[/tex] will be shifted to [tex]\( y = 2 \)[/tex].
### Step 4: Key Characteristics and Behavior of the Function
1. Asymptote: The graph will have a horizontal asymptote at [tex]\( y = 2 \)[/tex]. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^{x-1} \)[/tex] approaches 0, so [tex]\( f(x) \)[/tex] approaches 2.
2. Y-intercept: When [tex]\( x=0 \)[/tex]:
[tex]\[ f(0) = 2^{0-1} + 2 = 2^{-1} + 2 = \frac{1}{2} + 2 = 2.5 \][/tex]
3. Growth: Since it’s an exponential function with base greater than 1, it will grow rapidly as [tex]\( x \)[/tex] increases.
### Step 5: Sketching the Graph
- Horizontal asymptote at [tex]\( y = 2 \)[/tex].
- Y-intercept at [tex]\( (0, 2.5) \)[/tex].
- For [tex]\( x > 1 \)[/tex], the function grows exponentially because [tex]\( 2^{x-1} \)[/tex] increases rapidly.
### Step 6: Plot a Few More Points for Verification
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2^{1-1} + 2 = 2^0 + 2 = 1 + 2 = 3 \][/tex]
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2^{-1-1} + 2 = 2^{-2} + 2 = \frac{1}{4} + 2 = 2.25 \][/tex]
### Summary of Important Points:
- Horizontal Asymptote: [tex]\( y = 2 \)[/tex].
- Y-intercept: [tex]\( (0, 2.5) \)[/tex].
- Point (1, 3).
- Point (-1, 2.25).
- Rapid Exponential Growth for [tex]\( x > 1 \)[/tex].
Based on these characteristics, you should now be able to identify the correct graph among the given options. You are looking for a graph that:
- Approaches but never touches [tex]\( y = 2 \)[/tex] for negative [tex]\( x \)[/tex] values.
- Passes through the point [tex]\( (0, 2.5) \)[/tex].
- Passes through the point [tex]\( (1, 3) \)[/tex].
- Exhibits rapid exponential growth as [tex]\( x \)[/tex] increases.
Look for these features in the provided graph options to select the correct graph for [tex]\( f(x) = 2^{x-1} + 2 \)[/tex].
