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### Changes in Period and Phase Shift of Sine and Cosine Functions

Quiz

What is the general equation of a cosine function with an amplitude of 3, a period of [tex]$4 \pi$[/tex], and a horizontal shift of [tex]$-\pi$[/tex]?

A. [tex]$y = 4 \pi \cos(3(x - \pi))$[/tex]
B. [tex][tex]$y = 3 \cos(4 \pi(x + \pi))$[/tex][/tex]
C. [tex]$y = 3 \cos(0.5(x + \pi))$[/tex]
D. [tex]$y = 4 \pi \cos(0.2(x + \pi))$[/tex]



Answer :

GinnyAnswer
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.

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