Let's solve this step by step:
1. Given Values:
- Amplitude ([tex]\(A\)[/tex]): 3
- Period ([tex]\(T\)[/tex]): [tex]\(4\pi\)[/tex]
- Horizontal Shift ([tex]\(C\)[/tex]): [tex]\(-\pi\)[/tex]
2. General Equation of a Cosine Function:
The general form of a cosine function is:
[tex]\[
y = A \cos(B(x - C))
\][/tex]
where:
- [tex]\(A\)[/tex] is the amplitude,
- [tex]\(B\)[/tex] affects the period,
- [tex]\(C\)[/tex] is the horizontal shift.
3. Determine [tex]\(B\)[/tex]:
The period [tex]\(T\)[/tex] of a cosine function is given by:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
We need to solve for [tex]\(B\)[/tex]:
[tex]\[
4\pi = \frac{2\pi}{B}
\][/tex]
Multiply both sides by [tex]\(B\)[/tex]:
[tex]\[
4\pi B = 2\pi
\][/tex]
Divide both sides by [tex]\(4\pi\)[/tex]:
[tex]\[
B = \frac{2\pi}{4\pi} = \frac{1}{2} = 0.5
\][/tex]
4. Form the Equation:
Now, substituting the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
[tex]\[
y = 3 \cos\left(0.5 \left(x - (-\pi)\right)\right)
\][/tex]
Simplifying inside the cosine function:
[tex]\[
y = 3 \cos\left(0.5 (x + \pi)\right)
\][/tex]
5. Identify the Correct Option:
The correct option that matches this equation is:
[tex]\[
y = 3 \cos (0.5(x + \pi))
\][/tex]
Therefore, the general equation of the cosine function with the given conditions is:
[tex]\[
y = 3 \cos (0.5(x + \pi))
\][/tex]
This corresponds to the third given option.
