Certainly! Let's derive the formula for the given function step-by-step.
We are given the function in the form:
[tex]\[
y = 4.7 \cos \left( \frac{2 \pi}{[?]} (x - \square) \right) + \square
\][/tex]
Let's break down the components of a general cosine function and match them with our given function:
1. Amplitude (A): The amplitude is the coefficient in front of the cosine function. In our given function, the amplitude is 4.7.
2. Period (T): The period of a cosine function is given by [tex]\(\frac{2\pi}{B}\)[/tex], where [tex]\(B\)[/tex] is the coefficient multiplying [tex]\(x\)[/tex] inside the cosine function. In our function, [tex]\(B = \frac{2\pi}{[?]}\)[/tex], which means the period [tex]\(T = [?]\)[/tex].
3. Phase Shift (C): The phase shift is determined by the term [tex]\((x - C)\)[/tex] inside the cosine function. Here, it's [tex]\((x - \square)\)[/tex], implying the phase shift [tex]\( \phi \)[/tex] is [tex]\(\square\)[/tex].
4. Vertical Shift (D): The vertical shift is the constant term added to the cosine function. In our function, the vertical shift is [tex]\(\square\)[/tex].
With these components, we can express the function in the general form:
[tex]\[
y = A \cos \left( \frac{2 \pi}{T} (x - \phi) \right) + D
\][/tex]
For our specific function:
- [tex]\( A = 4.7 \)[/tex]
- [tex]\( \frac{2 \pi}{T} \)[/tex] implies the period [tex]\(T\)[/tex]
- [tex]\( \phi \)[/tex] represents the phase shift
- [tex]\( D \)[/tex] represents the vertical shift
Substituting these into the general form, we get:
[tex]\[
y = 4.7 \cos \left( \frac{2 \pi}{T} (x - \phi) \right) + D
\][/tex]
So, the formula for the given function is:
[tex]\[
y = 4.7 \cos \left( \frac{2 \pi}{T} (x - \phi) \right) + D
\][/tex]
In summary:
- The amplitude is 4.7.
- The period is denoted by [tex]\(T\)[/tex].
- The phase shift is denoted by [tex]\(\phi\)[/tex].
- The vertical shift is denoted by [tex]\(D\)[/tex].
Thus, the complete formula for the function is:
[tex]\[
y = 4.7 \cos \left( \frac{2 \pi}{T}(x - \phi) \right) + D
\][/tex]
