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Instructions: For the following function, select all the types of transformations that are occurring from the original function [tex]\(f(x) = 6^x\)[/tex].

Function: [tex]\(f(x) = \frac{1}{6}(4)^{2x} - 4\)[/tex]

Select one or more:
- Vertical Stretch
- Vertical Compression
- Vertical Reflection (over [tex]\(x\)[/tex]-axis)
- Vertical Shift Up
- Vertical Shift Down
- Horizontal Stretch
- Horizontal Compression
- Horizontal Reflection (over [tex]\(y\)[/tex]-axis)
- Horizontal Shift Right
- Horizontal Shift Left

Sagot :

GinnyAnswer
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
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