To write a cosine function with the given specifications, we need to use the general form of a cosine function:
[tex]\[ f(x) = A \cos(B(x - C)) + D \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude.
- [tex]\( B \)[/tex] is the frequency factor related to the period.
- [tex]\( C \)[/tex] is the horizontal shift.
- [tex]\( D \)[/tex] is the vertical shift or the midline.
Given:
- The amplitude [tex]\( A \)[/tex] is 4.
- The midline [tex]\( y = D \)[/tex] is 2.
- The period [tex]\( T \)[/tex] is [tex]\( \frac{5\pi}{2} \)[/tex].
To find [tex]\( B \)[/tex], we use the relationship between the period and [tex]\( B \)[/tex]:
[tex]\[ B = \frac{2\pi}{T} \][/tex]
Substituting the given period [tex]\( T \)[/tex]:
[tex]\[ B = \frac{2\pi}{\frac{5\pi}{2}} = \frac{2\pi \cdot 2}{5\pi} = \frac{4}{5} \][/tex]
Next, since we are not given any horizontal shift, we can assume [tex]\( C = 0 \)[/tex].
Now we can put everything into the general form of the cosine function:
[tex]\[ f(x) = 4 \cos\left(\frac{4}{5}x\right) + 2 \][/tex]
So, the cosine function is:
[tex]\[ f(x) = 4 \cos\left(\frac{4}{5}x\right) + 2 \][/tex]
Make sure to double-check that each component (amplitude, midline, and period) is correctly used and aligns with the given conditions.
