Question
Write the equation for a cosine function with amplitude \( 4 \pi \), period 2 , and phase shift -3 .
Ask by Martin Evans. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation for the cosine function is \( y = 4\pi \cos(\pi(x + 3)) \).
Solution
The standard form for a cosine function is
\[
y = A \cos\left(B(x - C)\right),
\]
where:
- \( A \) is the amplitude.
- The period \( T \) is given by \( T = \frac{2\pi}{B} \).
- \( C \) is the phase shift.
Given:
- Amplitude \( A = 4\pi \).
- Period \( T = 2 \).
- Phase shift \( C = -3 \) (which means the function is shifted to the left by 3 units).
First, find \( B \) using the period formula:
\[
T = \frac{2\pi}{B} \quad \Longrightarrow \quad 2 = \frac{2\pi}{B} \quad \Longrightarrow \quad B = \frac{2\pi}{2} = \pi.
\]
Now substitute \( A \), \( B \), and \( C \) into the standard form. Remember that a phase shift of \(-3\) means \( C = -3 \), so the expression inside the cosine becomes
\[
\pi\left(x - (-3)\right) = \pi(x + 3).
\]
Thus, the equation is
\[
y = 4\pi \cos\left(\pi(x + 3)\right).
\]
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Simplify this solution Beyond the Answer
The equation for a cosine function can be written in the form: \[ y = A \cos(B(x - C)) + D \] Where: - \( A \) is the amplitude, - \( B \) determines the period, - \( C \) is the phase shift, - \( D \) is the vertical shift (which we won't need in this specific case). Given the amplitude of \( 4\pi \), we have \( A = 4\pi \). The period is \( 2 \), and we calculate \( B \) using the formula \( \text{Period} = \frac{2\pi}{B} \). Solving for \( B \) gives us \( B = \frac{2\pi}{2} = \pi \). The phase shift of \( -3 \) means \( C = -3 \). Putting it all together, the equation is: \[ y = 4\pi \cos(\pi(x + 3)) \]
