Problem 20 - A Graphical Approach to Precalculus with Limits

Question

March each function defined in Column I with the appropriate description in Column \(\mathrm{II.}\) Do not use a calculator. I $$y=2 \sin (4 x-3)$$ II A. Amplitude \(=2 ;\) period \(=\frac{\pi}{2} ;\) phase shift \(=\frac{3}{4}\) B. Amplitude \(=3 ;\) period \(=\pi ;\) phase shift \(=2\) C. Amplitude \(=4 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{2}{3}\) D. Amplitude \(=2 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{4}{3}\)

Step-by-Step Solution

Answer

The function matches with description A.

Step 1: Identify Amplitude

The amplitude of a function in the form \(y = A \sin(Bx - C)\) is given by the absolute value of \(A\). In the function \(y = 2 \sin(4x - 3)\), the amplitude \(A = 2\).

Step 2: Calculate the Period

The period of a sine function \(y = A \sin(Bx - C)\) is given by \(\frac{2\pi}{B}\). For our function, \(B = 4\), so the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).

Step 3: Determine the Phase Shift

The phase shift of the function is calculated by \(\frac{C}{B}\) where \(y = A \sin(Bx - C)\). For \(Cx - 3\), \(C = 3\) and \(B = 4\). Thus, the phase shift is \(\frac{3}{4}\).

Step 4: Match with Description in Column II

Now, compare the calculated amplitude, period, and phase shift with the options provided in Column II. The correct match is description A: Amplitude \( = 2\); period \(=\frac{\pi}{2}\); phase shift \(=\frac{3}{4}\).

Key Concepts

Amplitude

When working with trigonometric functions, especially sine and cosine functions, the amplitude is a key feature. It tells us how "tall" a wave reaches from its midline. For the general form of a trigonometric function, such as \(y = A \sin(Bx - C)\), the amplitude is denoted by \(|A|\). This is the absolute value of the coefficient \(A\) in front of the sine or cosine.

To fully understand, think of amplitude as the maximum height from the center of the wave, or how far it reaches in either direction from the mid-point or axis of symmetry. In the function given in our exercise, \(y = 2 \sin(4x - 3)\), the amplitude is \(2\). This tells us that the highest and lowest points of the sine wave are 2 units away from the middle. It helps predict the peak height and depth of the wave. Thus, correctly identifying the amplitude is essential in understanding the overall behavior of the trigonometric function.

Period

The period of a trigonometric function indicates the interval it takes for the function to complete one full cycle of its wave. For sine and cosine functions described by \(y = A \sin(Bx - C)\), the period is calculated by \(\frac{2\pi}{B}\).

The concept of the period helps us determine how "stretched" or "compressed" the wave looks horizontally. If \(B\) is large, the function repeats more frequently, leading to a shorter period; if \(B\) is small, the function stretches further, resulting in a longer period. Here, in the function \(y = 2 \sin(4x - 3)\), \(B = 4\). Thus, its period is \(\frac{2\pi}{4} = \frac{\pi}{2}\). This value tells us that every \(\frac{\pi}{2}\) units along the x-axis, the wave pattern of the sine function starts anew. It plays a crucial role in understanding the function's wave behavior and how often it "returns to normal" over the x-values.

Phase Shift

The phase shift of a trigonometric function tells us how the graph of the function "slides" horizontally along the x-axis. Mathematically, for a function in the form \(y = A \sin(Bx - C)\), the phase shift is computed as \(\frac{C}{B}\).

Phase shift is crucial because it alters the starting point of the sine or cosine wave. If the phase shift is positive, the graph moves to the right; if negative, it shifts to the left. This movement adjusts where the repeating wave pattern begins in relation to the origin. For the function \(y = 2 \sin(4x - 3)\), using the values \(C = 3\) and \(B = 4\), the phase shift calculates to \(\frac{3}{4}\).

This means our sine wave starts \(\frac{3}{4}\) units to the right. Understanding phase shift helps in examining how the wave's starting place is affected, further aiding in graphing and analyzing the behavior of the function across different x-values.