Question
Vertical compression and stretch
Identify the following transformation
Y=-3(4)[^x+3]+2 arrow y=-6(4)^[x+6]+4
Identify the following transformation
Y=-3(4)[^x+3]+2 arrow y=-6(4)^[x+6]+4
Answers
GPT-4o mini
To analyze the transformation from the equation \( y = -3(4)^{x+3} + 2 \) to \( y = -6(4)^{x+6} + 4 \), we will break down the changes in each component of the equation.
1. **Vertical Stretch/Compression and Reflection**:
- The coefficient of \( 4^x \) changed from \(-3\) to \(-6\).
- This indicates a vertical stretch by a factor of 2 (since \(-6\) is twice \(-3\)) and a reflection across the x-axis (the negative sign).
2. **Horizontal Shift**:
- The exponent in the first equation has \( x + 3 \) and in the second it has \( x + 6 \).
- This indicates a horizontal shift. From \( x + 3 \) to \( x + 6 \) represents a shift to the left by 3 units (from \( -3 \) to \( -6 \)).
3. **Vertical Shift**:
- The constant term changed from \( +2 \) to \( +4 \).
- This represents a vertical shift upwards by 2 units (from \( 2 \) to \( 4 \)).
### Summary of Transformations:
- **Vertical Reflection**: The negative outside the function indicates a reflection across the x-axis.
- **Vertical Stretch**: The factor changes from \(-3\) to \(-6\), indicating a vertical stretch by a factor of 2.
- **Horizontal Shift**: The entire graph is shifted to the left by 3 units.
- **Vertical Shift**: The graph is shifted upwards by 2 units.
In conclusion, the transformation includes a vertical reflection, a vertical stretch, a horizontal shift to the left, and a vertical shift upwards.
1. **Vertical Stretch/Compression and Reflection**:
- The coefficient of \( 4^x \) changed from \(-3\) to \(-6\).
- This indicates a vertical stretch by a factor of 2 (since \(-6\) is twice \(-3\)) and a reflection across the x-axis (the negative sign).
2. **Horizontal Shift**:
- The exponent in the first equation has \( x + 3 \) and in the second it has \( x + 6 \).
- This indicates a horizontal shift. From \( x + 3 \) to \( x + 6 \) represents a shift to the left by 3 units (from \( -3 \) to \( -6 \)).
3. **Vertical Shift**:
- The constant term changed from \( +2 \) to \( +4 \).
- This represents a vertical shift upwards by 2 units (from \( 2 \) to \( 4 \)).
### Summary of Transformations:
- **Vertical Reflection**: The negative outside the function indicates a reflection across the x-axis.
- **Vertical Stretch**: The factor changes from \(-3\) to \(-6\), indicating a vertical stretch by a factor of 2.
- **Horizontal Shift**: The entire graph is shifted to the left by 3 units.
- **Vertical Shift**: The graph is shifted upwards by 2 units.
In conclusion, the transformation includes a vertical reflection, a vertical stretch, a horizontal shift to the left, and a vertical shift upwards.