To transform the parent cosine function [tex]\( f(x) = \cos(x) \)[/tex] into the function [tex]\( d(x) = \cos(2x - 1) + 5 \)[/tex], we need to identify and describe the series of transformations applied. Here are the detailed steps for each transformation:
1. Horizontal Shift:
- Inside the cosine function, we have [tex]\(2x - 1\)[/tex]. To determine the horizontal shift, we set the inner expression equal to zero:
[tex]\[
2x - 1 = 0 \implies x = \frac{1}{2}
\][/tex]
- This indicates that the graph is shifted to the right by [tex]\(\frac{1}{2}\)[/tex] units.
2. Vertical Shift:
- The constant [tex]\(+5\)[/tex] outside the cosine function shifts the graph vertically.
- This means the graph of the cosine function is shifted up by 5 units.
3. Frequency:
- The coefficient 2 inside the cosine function affects the frequency. The usual period of the cosine function is [tex]\(2\pi\)[/tex]. With a coefficient of 2, the frequency is doubled, and the period is halved:
[tex]\[
\text{Frequency} = 2
\][/tex]
- Therefore, the cosine function oscillates more frequently.
So, the transformations applied to the parent cosine function [tex]\( f(x) = \cos(x) \)[/tex] to create [tex]\( d(x) = \cos(2x - 1) + 5 \)[/tex] are:
- Horizontal shift: [tex]\(\frac{1}{2}\)[/tex] units to the right.
- Vertical shift: 5 units upwards.
- Frequency: doubled (frequency = 2).
Therefore, the correct answers from the drop-down menu should be:
- Horizontal shift: [tex]\( \frac{1}{2} \)[/tex] units to the right.
- Vertical shift: 5 units upwards.
- Frequency: 2.
