To find the formula for the given function, let's break it down step-by-step.
The given function is of the form:
[tex]\[ y = 1.7 \cos \left(\frac{2 \pi}{[?]}(x - \square)\right) + \ldots \][/tex]
This form is a cosine function [tex]\( y = A \cos \left(B (x - C)\right) + D \)[/tex], where [tex]\( A \)[/tex] is the amplitude, [tex]\( B \)[/tex] is the frequency, [tex]\( C \)[/tex] is the phase shift, and [tex]\( D \)[/tex] is the vertical shift.
1. Identify the Amplitude (A):
The amplitude [tex]\( A \)[/tex] is the coefficient in front of the cosine function. In this case:
[tex]\[ A = 1.7 \][/tex]
2. Determine the Frequency (B) and the Period (T):
For a cosine function of the form [tex]\( \cos(B (x - C)) \)[/tex], [tex]\( B \)[/tex] is related to the period [tex]\( T \)[/tex] by:
[tex]\[ B = \frac{2 \pi}{T} \][/tex]
Given that [tex]\( B = \frac{2 \pi}{[?]} \)[/tex], let’s consider that the period [tex]\( T \)[/tex] is determined as follows:
[tex]\[ B = 2 \pi \][/tex]
Therefore,
[tex]\[ T = \frac{2 \pi}{2 \pi} = 1 \][/tex]
3. Identify the Phase Shift (C):
The phase shift [tex]\( C \)[/tex] in the expression [tex]\( \left(x - C\right) \)[/tex] is given directly. Here:
[tex]\[ C = 0 \][/tex]
This implies that there is no horizontal shift in the function.
4. Construct the Function:
Plugging in [tex]\( A = 1.7 \)[/tex], [tex]\( B = 2\pi \)[/tex], and [tex]\( C = 0 \)[/tex], the complete function becomes:
[tex]\[ y = 1.7 \cos (2\pi (x - 0)) \][/tex]
5. Simplify the Equation:
Since [tex]\( C = 0 \)[/tex]:
[tex]\[ y = 1.7 \cos (2\pi x) \][/tex]
So, the formula for the function is:
[tex]\[
y = 1.7 \cos (6.283185307179586 \cdot (x - 0))
\][/tex]
This expression simplifies to:
[tex]\[
y = 1.7 \cos (6.283185307179586 x)
\][/tex]
