To identify the transformations of the function [tex]\( f(x) = \frac{1}{6}(4)^{2x} - 4 \)[/tex] from the original function [tex]\( f(x) = 6^x \)[/tex], we need to analyze each part of the transformation.
1. Vertical Compression:
The coefficient [tex]\(\frac{1}{6}\)[/tex] in front of the exponential term indicates a vertical compression by a factor of [tex]\( \frac{1}{6} \)[/tex].
2. Horizontal Compression:
The exponent [tex]\(2x\)[/tex] inside the exponential function means that there is a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex]. This is because the factor inside the exponent effectively "speeds up" the growth of the function, compressing it horizontally.
3. Vertical Shift Down:
The term [tex]\(-4\)[/tex] at the end of the function indicates a vertical shift down by 4 units.
Therefore, the types of transformations are:
- Vertical Compression
- Horizontal Compression
- Vertical Shift Down
So the correct transformations to select are:
- Vertical Compression
- Horizontal Compression
- Vertical Shift Down
