Certainly! To find a formula for the given trigonometric function, let's start with the general form of a cosine function with given amplitude, period, and phase shift.
The general form of a cosine function is:
[tex]\[
y = A \cos\left(\frac{2\pi}{T} (x - P)\right) + C
\][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude of the function,
- [tex]\( T \)[/tex] is the period of the function,
- [tex]\( P \)[/tex] is the phase shift (horizontal shift) of the function,
- [tex]\( C \)[/tex] is the vertical shift of the function.
Given the expression:
[tex]\[
y = 1.9 \cos\left(\frac{2 \pi}{[?]}(x - \quad)\right) + 0
\][/tex]
we need to identify and match the corresponding parameters:
1. Amplitude (A):
The coefficient of the cosine function gives the amplitude [tex]\( A \)[/tex]. Here, it is clearly [tex]\( 1.9 \)[/tex].
2. Vertical Shift (C):
The constant added to the cosine function gives the vertical shift [tex]\( C \)[/tex]. Here, it is [tex]\( 0 \)[/tex], indicating no vertical shift.
3. Period (T):
The period [tex]\( T \)[/tex] is given inside the argument of the cosine function. The expression [tex]\( \frac{2 \pi}{T} \)[/tex] represents how frequently the function completes one full cycle.
Since the period value is not given explicitly, we denote it as [tex]\( T \)[/tex].
4. Phase Shift (P):
The phase shift [tex]\( P \)[/tex] is represented by the horizontal shift [tex]\( x - P \)[/tex]. Similarly, since the phase shift is not specified in the expression, we denote it as [tex]\( P \)[/tex].
Putting it all together, we have:
[tex]\[
y = 1.9 \cos\left(\frac{2 \pi}{T} (x - P)\right) + 0
\][/tex]
This formula represents a cosine function with an amplitude of [tex]\( 1.9 \)[/tex], no vertical shift ([tex]\( C = 0 \)[/tex]), an unspecified period [tex]\( T \)[/tex], and an unspecified phase shift [tex]\( P \)[/tex].
Therefore, the function formula in terms of [tex]\( T \)[/tex] and [tex]\( P \)[/tex] is:
[tex]\[
y = 1.9 \cos\left(\frac{2 \pi}{T} (x - P)\right) + 0
\][/tex]
This is the desired result for the function.
