To graph the function [tex]\( f(x) = -0.1 \cos(x) - 4 \)[/tex] from the parent cosine function [tex]\( \cos(x) \)[/tex], let's break down the specific transformations involved:
1. Reflection Across the [tex]\( x \)[/tex]-axis:
- The negative sign in front of the cosine function, [tex]\( -0.1 \cos(x) \)[/tex], indicates a reflection across the [tex]\( x \)[/tex]-axis.
- Graphically, this flips the graph of [tex]\( \cos(x) \)[/tex] over the [tex]\( x \)[/tex]-axis.
2. Vertical Compression by a Factor of 0.1:
- The coefficient [tex]\( 0.1 \)[/tex] (which is less than 1 and positive) in [tex]\( -0.1 \cos(x) \)[/tex] represents a vertical compression.
- This means that the amplitude of the cosine function is reduced by a factor of 0.1, making the peaks and valleys of the wave closer to the [tex]\( x \)[/tex]-axis.
3. Vertical Translation Down by 4 Units:
- The [tex]\( -4 \)[/tex] at the end of the function, [tex]\( -0.1 \cos(x) - 4 \)[/tex], shifts the entire graph down by 4 units.
- This means every point on the graph of [tex]\( -0.1 \cos(x) \)[/tex] is moved 4 units lower on the [tex]\( y \)[/tex]-axis.
Putting all these transformations together, the set of transformations needed are:
- Reflection across the [tex]\( x \)[/tex]-axis
- Vertical compression by a factor of 0.1
- Vertical translation 4 units down
Therefore, the correct set of transformations is:
- Reflection across the [tex]\( x \)[/tex]-axis
- Vertical compression by a factor of 0.1
- Vertical translation 4 units down
Thus, the correct choice is:
Reflection across the [tex]\( x \)[/tex]-axis, vertical compression by a factor of 0.1, vertical translation 4 units down
