Sure, let's create a cosine function that meets the following criteria: an amplitude of 4, a midline of [tex]\( y = 5 \)[/tex], and a period of [tex]\( \frac{1}{4} \)[/tex].
1. Amplitude (A):
The amplitude of a cosine function is the coefficient in front of the cosine term. It determines the maximum and minimum values the function reaches from the midline. Given the amplitude is 4, we have:
[tex]\[
A = 4
\][/tex]
2. Midline (D):
The midline is the horizontal line that the function oscillates around. In a cosine function of the form [tex]\( y = A \cos(B(x - C)) + D \)[/tex], [tex]\( D \)[/tex] represents the midline. Given the midline is [tex]\( y = 5 \)[/tex], we have:
[tex]\[
D = 5
\][/tex]
3. Period:
The period of a cosine function is the distance over which the function completes one full cycle. The period [tex]\( T \)[/tex] is related to the coefficient [tex]\( B \)[/tex] inside the cosine function by the formula:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
Given the period is [tex]\( \frac{1}{4} \)[/tex], we solve for [tex]\( B \)[/tex]:
[tex]\[
\frac{1}{4} = \frac{2\pi}{B} \implies B = 8\pi
\][/tex]
Putting it all together, we now have the values of [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( D \)[/tex]. There is no horizontal shift [tex]\( C \)[/tex] specified, so we assume it to be 0.
The cosine function that meets these criteria is:
[tex]\[
y = 4 \cos(8\pi x) + 5
\][/tex]
Therefore, the cosine function is:
[tex]\[
y = 4 \cos(8\pi x) + 5
\][/tex]
