To identify the transformations in each function, we'll analyze how each function is related to the parent function, which is \( f(x) = \log_6 x \).
1. \( a(x) = 3 \log_6 x \)
- Transformation: Vertical Stretch
- Explanation: The 3 in front of the logarithm indicates a vertical stretch by a factor of 3.
2. \( b(x) = \log_6 x + 3 \)
- Transformation: Vertical Shift
- Explanation: The +3 at the end of the function indicates a vertical shift upward by 3 units.
3. \( c(x) = \log_6 3x \)
- Transformation: Horizontal Shrink
- Explanation: This can be rewritten using logarithmic properties: \( c(x) = \log_6 3 + \log_6 x \). The term \( 3x \) can be understood as \( 3 \) (a constant multiplier) inside the logarithm, indicating a horizontal shrink (by a factor of \( \frac{1}{3} \)).
4. \( d(x) = \log_6(x + 3) \)
- Transformation: Horizontal Shift
- Explanation: The term \( x + 3 \) indicates a horizontal shift to the left by 3 units, since we replace \( x \) with \( x - (-3) \).
To summarize:
- \( a(x) = 3 \log_6 x \): Vertical Stretch
- \( b(x) = \log_6 x + 3 \): Vertical Shift (upward by 3)
- \( c(x) = \log_6 3x \): Horizontal Shrink
- \( d(x) = \log_6(x + 3) \): Horizontal Shift (left by 3)