Question
describe the transformation of this function: y = -2 (3)^(x+3), from y = 3^x
Answers
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GPT-4o mini
To understand the transformation of the function \( y = -2(3)^{(x+3)} \) from the base function \( y = 3^x \), we can break it down into several steps. Here's how the transformations work:
1. **Horizontal Shift**:
The term \( (x + 3) \) in the exponent indicates a horizontal shift. Specifically, \( y = 3^{(x + 3)} \) represents a shift of the graph of \( y = 3^x \) to the left by 3 units.
2. **Vertical Stretch and Reflection**:
The coefficient \(-2\) in front of the function signifies two transformations:
- The negative sign indicates a reflection about the x-axis. This means the graph will flip upside down compared to the original function \( y = 3^x \).
- The factor of 2 is a vertical stretch. This will stretch the graph away from the x-axis by a factor of 2. Hence, the points on the original graph will be stretched vertically, making them twice as far from the x-axis.
Combining these transformations, we can summarize:
- Start with the parent function \( y = 3^x \).
- Shift the graph 3 units to the left to get \( y = 3^{(x + 3)} \).
- Reflect the graph across the x-axis to convert it to \( y = -3^{(x + 3)} \).
- Stretch it vertically by a factor of 2, resulting in \( y = -2(3^{(x + 3)}) \).
Thus, the final transformed function \( y = -2(3)^{(x + 3)} \) is a reflection of the original function that has been shifted 3 units to the left and stretched vertically by a factor of 2.
1. **Horizontal Shift**:
The term \( (x + 3) \) in the exponent indicates a horizontal shift. Specifically, \( y = 3^{(x + 3)} \) represents a shift of the graph of \( y = 3^x \) to the left by 3 units.
2. **Vertical Stretch and Reflection**:
The coefficient \(-2\) in front of the function signifies two transformations:
- The negative sign indicates a reflection about the x-axis. This means the graph will flip upside down compared to the original function \( y = 3^x \).
- The factor of 2 is a vertical stretch. This will stretch the graph away from the x-axis by a factor of 2. Hence, the points on the original graph will be stretched vertically, making them twice as far from the x-axis.
Combining these transformations, we can summarize:
- Start with the parent function \( y = 3^x \).
- Shift the graph 3 units to the left to get \( y = 3^{(x + 3)} \).
- Reflect the graph across the x-axis to convert it to \( y = -3^{(x + 3)} \).
- Stretch it vertically by a factor of 2, resulting in \( y = -2(3^{(x + 3)}) \).
Thus, the final transformed function \( y = -2(3)^{(x + 3)} \) is a reflection of the original function that has been shifted 3 units to the left and stretched vertically by a factor of 2.