Problem 1 - Contemporary Precalculus

Question

State the amplitude, period, and phase shift of the function. \(g(t)=3 \sin (2 t-\pi)\)

Step-by-Step Solution

Answer

Answer: The amplitude is 3, the period is \(\pi\), and the phase shift is \(\frac{\pi}{2}\).

Step 1: Identify the amplitude

The amplitude of a sinusoidal function is the absolute value of the constant \(A\). In our given function \(g(t) = 3 \sin (2t - \pi)\), the constant \(A\) is 3. Thus, the amplitude of the given function is \(|3| = 3\).

Step 2: Calculate the period

The period of a sinusoidal function is calculated by the formula \(\text{Period} = \frac{2\pi}{|B|}\). In our given function \(g(t) = 3 \sin (2t - \pi)\), the constant \(B\) is 2. To find the period, plug the value of \(B\) into the formula: \(\text{Period} = \frac{2\pi}{|2|} = \frac{2\pi}{2} = \pi\). So, the period of the given function is \(\pi\).

Step 3: Find the phase shift

The phase shift of a sinusoidal function can be determined by the constant \(C\). In our given function \(g(t) = 3 \sin (2t - \pi)\), the constant \(C\) is \(\pi\). The phase shift is calculated using the formula \(\text{Phase Shift} = \frac{C}{B}\). Plug in the values of \(B\) and \(C\): \(\text{Phase Shift} = \frac{\pi}{2} = \frac{\pi}{2}\). So, the phase shift of the given function is \(\frac{\pi}{2}\). To summarize: - Amplitude: 3 - Period: \(\pi\) - Phase Shift: \(\frac{\pi}{2}\)

Key Concepts

Amplitude

In trigonometric functions, the amplitude represents the peak value of the wave from its central axis. Think of it as how tall the wave is from the middle point both upwards and downwards. It gives us a measure of the wave's strength or intensity. For sinusoidal functions, which include sine (\( \sin \) and cosine (\( \cos \) functions, the amplitude is determined by the multiplier of the trigonometric function.

• In the case of the function \( g(t) = 3 \sin(2t - \pi) \), the amplitude is found from the coefficient of the \( \sin \) function, which is 3.
• Amplitudes are always positive, so we consider the absolute value: \( |3| = 3 \).

Understanding amplitude is essential as it informs you how far the function will oscillate above and below its average value. So, in simple terms, in our example, the wave is 3 units above and below the mid-line.

Period

The period of a trigonometric function is the interval over which the function completes one full wave cycle before repeating itself. Think of it as the duration of one cycle of the wave.For functions like sine and cosine, the period is calculated using the formula: \[ \text{Period} = \frac{2\pi}{|B|} \]where \( B \) is the coefficient of the variable inside the function (in our case, it's attached to \( t \)).

• For our function \( g(t) = 3 \sin(2t - \pi) \), the value of \( B \) is 2.
• Applying the formula, the period becomes \( \frac{2\pi}{2} = \pi \).

This means the wave pattern of our function repeats itself every \( \pi \) units along the horizontal axis. Recognizing the period is crucial for predicting the function's behavior over time or distance, making it an important concept in both mathematics and applied sciences.

Phase Shift

Phase shift refers to the horizontal movement or "shift" in the wave on the graph. It tells you where the wave begins its cycle. Utilized in adjusting wave functions horizontally, the phase shift is determined by the formula: \[ \text{Phase Shift} = \frac{C}{B} \]where \( C \) is the constant term subtracted or added inside the angle of the function, and \( B \) is the coefficient of the variable.In our function,

• The expression inside the sine function is \( 2t - \pi \), with \( C = \pi \).
• The value of \( B \) is 2, leading to a phase shift of \( \frac{\pi}{2} \).

So our function \( g(t) \) is shifted \( \frac{\pi}{2} \) units to the right. Phase shifts are important because they indicate how far the wave has moved from its standard starting position. It's especially useful in analyzing functionalities in electronics, sound waves, and other periodic phenomena.