Question
Describe the transformation of the graph of f(x)=2^x to graph option 3
Crosses y axis at (0,1) points (-1,2), (-2,4), (1,0.5), then almost touching x axis to (6,0)
Options
-f(x)
f(-x)
-f(-x)
f(x-1)
Crosses y axis at (0,1) points (-1,2), (-2,4), (1,0.5), then almost touching x axis to (6,0)
Options
-f(x)
f(-x)
-f(-x)
f(x-1)
Answers
Answered by
GPT-4o mini
To understand the transformations of the graph of \( f(x) = 2^x \) to a new graph, let's first analyze the original function and the transformations that each option represents.
### Given Function:
- **Function**: \( f(x) = 2^x \)
- **Key Characteristics**:
- Crosses the y-axis at (0, 1) because \( 2^0 = 1 \).
- For \( x = -1, f(-1) = 2^{-1} = 0.5 \).
- For \( x = -2, f(-2) = 2^{-2} = 0.25 \).
- For \( x = 1, f(1) = 2^1 = 2 \).
- As \( x \) approaches negative infinity, \( f(x) \) approaches 0 (the x-axis).
### Analyzing Each Option:
1. **Option: \( -f(x) \)**
- Transformation: Reflects the graph across the x-axis.
- Original points become: (0, -1), (-1, -2), (-2, -4), (1, -0.5), and it would approach the x-axis from below.
- **Does not match** the given transformation.
2. **Option: \( f(-x) \)**
- Transformation: Reflects the graph across the y-axis.
- Original points become: (0, 1), (1, 2), (2, 4), (-1, 0.5), and it would still approach the x-axis asymptotically.
- **Does not match** the given transformation.
3. **Option: \( -f(-x) \)**
- Transformation: Reflects the graph across the y-axis and then across the x-axis.
- Original points become: (0, -1), (1, -2), (2, -4), (-1, -0.5), and it would approach the x-axis from below.
- **Does not match** the given transformation.
4. **Option: \( f(x-1) \)**
- Transformation: Shifts the graph to the right by 1 unit.
- For the transformed function:
- At \( x = 0 \): \( f(0-1) = f(-1) = 0.5 \) → point at (0, 0.5)
- At \( x = 1 \): \( f(1-1) = f(0) = 1 \) → point at (1, 1)
- At \( x = 2 \): \( f(2-1) = f(1) = 2 \) → point at (2, 2)
- At \( x = 3 \): \( f(3-1) = f(2) = 4 \) → point at (3, 4)
- At \( x = -1 \): \( f(-1-1) = f(-2) \approx 0.25 \) → point at (-1, 0.25), etc.
### Conclusion
The transformation of the graph leads to the points provided in the description:
- \( f(x) = 2^x \) transformed to mirror the described points can be related best to the option that shifts the graph to the right.
Thus, the correct answer that corresponds with the transformation you described — where points near the x-axis approach zero and other points are located at specific coordinates resembling the y-values mentioned — is **Option 4: \( f(x-1) \)**.
### Given Function:
- **Function**: \( f(x) = 2^x \)
- **Key Characteristics**:
- Crosses the y-axis at (0, 1) because \( 2^0 = 1 \).
- For \( x = -1, f(-1) = 2^{-1} = 0.5 \).
- For \( x = -2, f(-2) = 2^{-2} = 0.25 \).
- For \( x = 1, f(1) = 2^1 = 2 \).
- As \( x \) approaches negative infinity, \( f(x) \) approaches 0 (the x-axis).
### Analyzing Each Option:
1. **Option: \( -f(x) \)**
- Transformation: Reflects the graph across the x-axis.
- Original points become: (0, -1), (-1, -2), (-2, -4), (1, -0.5), and it would approach the x-axis from below.
- **Does not match** the given transformation.
2. **Option: \( f(-x) \)**
- Transformation: Reflects the graph across the y-axis.
- Original points become: (0, 1), (1, 2), (2, 4), (-1, 0.5), and it would still approach the x-axis asymptotically.
- **Does not match** the given transformation.
3. **Option: \( -f(-x) \)**
- Transformation: Reflects the graph across the y-axis and then across the x-axis.
- Original points become: (0, -1), (1, -2), (2, -4), (-1, -0.5), and it would approach the x-axis from below.
- **Does not match** the given transformation.
4. **Option: \( f(x-1) \)**
- Transformation: Shifts the graph to the right by 1 unit.
- For the transformed function:
- At \( x = 0 \): \( f(0-1) = f(-1) = 0.5 \) → point at (0, 0.5)
- At \( x = 1 \): \( f(1-1) = f(0) = 1 \) → point at (1, 1)
- At \( x = 2 \): \( f(2-1) = f(1) = 2 \) → point at (2, 2)
- At \( x = 3 \): \( f(3-1) = f(2) = 4 \) → point at (3, 4)
- At \( x = -1 \): \( f(-1-1) = f(-2) \approx 0.25 \) → point at (-1, 0.25), etc.
### Conclusion
The transformation of the graph leads to the points provided in the description:
- \( f(x) = 2^x \) transformed to mirror the described points can be related best to the option that shifts the graph to the right.
Thus, the correct answer that corresponds with the transformation you described — where points near the x-axis approach zero and other points are located at specific coordinates resembling the y-values mentioned — is **Option 4: \( f(x-1) \)**.