Question
g(x)=1/4 times3^(x-5)+8
describe completely all of the transformations that have been performed on the parent function, f(x)=3^x to give us the g(x) function above
describe completely all of the transformations that have been performed on the parent function, f(x)=3^x to give us the g(x) function above
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GPT-4o mini
To describe the transformations applied to the parent function \( f(x) = 3^x \) to obtain the function \( g(x) = \frac{1}{4} \cdot 3^{(x-5)} + 8 \), we will analyze each part of the transformation step by step.
1. **Horizontal Shift**: The term \( (x - 5) \) indicates a horizontal shift.
- Since \( g(x) \) has \( 3^{(x - 5)} \), this represents a shift to the right by 5 units.
- Thus, \( f(x) = 3^x \) becomes \( 3^{(x - 5)} \).
2. **Vertical Stretch/Compression**: The coefficient \( \frac{1}{4} \) in front of \( 3^{(x - 5)} \) indicates a vertical compression.
- Since the coefficient is \( \frac{1}{4} \), the function is compressed vertically by a factor of 4.
- This compresses the value of the function compared to \( 3^{(x-5)} \).
3. **Vertical Shift**: The addition of \( 8 \) at the end of the function indicates a vertical shift.
- Adding \( 8 \) shifts the entire graph upward by 8 units.
- Therefore, the entire function \( \frac{1}{4} \cdot 3^{(x - 5)} \) is moved up by 8 units.
Putting all these transformations together, we can summarize the transformations applied to the parent function \( f(x) = 3^x \) to arrive at \( g(x) \):
- **Shift right** by \( 5 \) units.
- **Vertical compression** by a factor of \( \frac{1}{4} \).
- **Shift up** by \( 8 \) units.
So the complete sequence of transformations is: Right shift 5 units, vertical compression by 4, then upward shift by 8 units.
1. **Horizontal Shift**: The term \( (x - 5) \) indicates a horizontal shift.
- Since \( g(x) \) has \( 3^{(x - 5)} \), this represents a shift to the right by 5 units.
- Thus, \( f(x) = 3^x \) becomes \( 3^{(x - 5)} \).
2. **Vertical Stretch/Compression**: The coefficient \( \frac{1}{4} \) in front of \( 3^{(x - 5)} \) indicates a vertical compression.
- Since the coefficient is \( \frac{1}{4} \), the function is compressed vertically by a factor of 4.
- This compresses the value of the function compared to \( 3^{(x-5)} \).
3. **Vertical Shift**: The addition of \( 8 \) at the end of the function indicates a vertical shift.
- Adding \( 8 \) shifts the entire graph upward by 8 units.
- Therefore, the entire function \( \frac{1}{4} \cdot 3^{(x - 5)} \) is moved up by 8 units.
Putting all these transformations together, we can summarize the transformations applied to the parent function \( f(x) = 3^x \) to arrive at \( g(x) \):
- **Shift right** by \( 5 \) units.
- **Vertical compression** by a factor of \( \frac{1}{4} \).
- **Shift up** by \( 8 \) units.
So the complete sequence of transformations is: Right shift 5 units, vertical compression by 4, then upward shift by 8 units.
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