Problem 110 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the amplitude, period, and phase shift of $$ y=\frac{3}{2} \cos (6 x+3 \pi) $$

Step-by-Step Solution

Answer

Amplitude: \( \frac{3}{2} \), Period: \( \frac{\pi}{3} \), Phase Shift: \( -\frac{\pi}{2} \).

Step 1: Identify the Coefficient of the Trigonometric Function

The given function is \( y = \frac{3}{2} \cos(6x + 3\pi) \). The coefficient in front of the cosine function is \( \frac{3}{2} \), which represents the amplitude.

Step 2: Determine the Amplitude

The amplitude of a trigonometric function \( y = A \cos(Bx + C) \) is the absolute value of \( A \). Hence, the amplitude is \( \left| \frac{3}{2} \right| = \frac{3}{2} \).

Step 3: Identify the Period

The period of a cosine function \( y = A \cos(Bx + C) \) is given by \( \frac{2\pi}{B} \). Here, \( B = 6 \).

Step 4: Calculate the Period

Substitute \( B \) into the formula to find the period: \( \frac{2\pi}{6} = \frac{\pi}{3} \). Therefore, the period is \( \frac{\pi}{3} \).

Step 5: Determine the Phase Shift

The phase shift of a cosine function \( y = A \cos(Bx + C) \) is given by \( \frac{-C}{B} \). Here, \( C = 3\pi \) and \( B = 6 \).

Step 6: Calculate the Phase Shift

Substitute \( C \) and \( B \) into the formula: \( \frac{-3\pi}{6} = -\frac{\pi}{2} \). Therefore, the phase shift is \( -\frac{\pi}{2} \).

Key Concepts

Amplitude Calculation

In trigonometric functions like the one given, identifying the amplitude is the first step. The amplitude represents the maximum value that the function can reach.
This value directly indicates how high and low the graph of the function will go. It is found by looking at the coefficient in front of the trigonometric function.

Given the function:\br>$$ y = \frac{3}{2} \, \text{cos} \, (6x + 3\pi) $$
You find the coefficient in front of the cosine, which is \frac{3}{2}\. The amplitude is the absolute value of this coefficient.
So, here:

• Coefficient = \(\frac{3}{2}\)
• Thus, Amplitude = \(| \frac{3}{2} |\) = \(\frac{3}{2}\)

This means that the function reaches a maximum height of \frac{3}{2} and a minimum of \(\frac{-3}{2}\). Understanding this is crucial to grasp how the function behaves.”\br> “\br>},\br> { \br>