Problem 49 - Precalculus

Question

In Exercises \(49-60\), state the amplitude, period, and phase shift (including direction) of the given function. $$y=2 \sin (\pi x-1)$$

Step-by-Step Solution

Answer

Amplitude: 2, Period: 2, Phase shift: \(\frac{1}{\pi}\) to the right.

Step 1: Identify the General Form of the Sine Function

The general form of the sine function is \( y = a \sin(bx - c) \), where:- \(a\) is the amplitude,- \(b\) affects the period,- \(c\) affects the phase shift. The given function is \( y = 2 \sin(\pi x - 1) \). Here, \( a = 2 \), \( b = \pi \), and \( c = 1 \).

Step 2: Determine the Amplitude

The amplitude of a sine function \(y = a \sin(bx - c)\) is the absolute value of \(a\). In this function, \(a = 2\). Therefore, the amplitude is given by:\[|a| = |2| = 2\]

Step 3: Calculate the Period

The period of a sine function \(y = a \sin(bx - c)\) is calculated by the formula:\[\text{Period} = \frac{2\pi}{b}\]For the given function, \(b = \pi\), so:\[\text{Period} = \frac{2\pi}{\pi} = 2\]

Step 4: Determine the Phase Shift

The phase shift of the sine function \(y = a \sin(bx - c)\) can be found using the formula:\[\text{Phase Shift} = \frac{c}{b}\]Substituting \(c = 1\) and \(b = \pi\) into the formula gives:\[\text{Phase Shift} = \frac{1}{\pi}\]This is a shift to the right, as the expression \(bx - c\) suggests a shift in the positive x-direction.

Key Concepts

Amplitude

The amplitude of a trigonometric function like sine or cosine tells us how far the graph stretches vertically. It's essentially the height of the wave from the center line to the peak. For any sine function given by the equation \(y = a \sin(bx - c)\), the amplitude is represented by the absolute value of \(a\). This means we are only concerned with the magnitude, not the direction, as amplitude is always positive.

In our specific example, \(y = 2 \sin(\pi x - 1)\), the amplitude is determined by \(a = 2\). The absolute value \(|2| = 2\) tells us that the height of the sine wave from its midline to either a peak or a trough is 2 units. This impacts how tall the wave appears when plotted, but does not affect the horizontal stretches or shifts.

Period

The period of a sine or cosine function defines how long it takes for the graph of the function to complete one full cycle and start repeating itself. This is equivalent to the distance along the x-axis before the wave pattern begins anew. For the function \(y = a \sin(bx - c)\), the period is calculated using the formula \(\frac{2\pi}{b}\).

In the case of the function \(y = 2 \sin(\pi x - 1)\), the value \(b = \pi\). Plugging this into our formula gives us:

• \(\text{Period} = \frac{2\pi}{\pi} = 2\)

Thus, the function completes one full wave cycle every 2 units along the x-axis. A smaller period indicates a quicker oscillation of the wave, while a larger period indicates a slower oscillation.

Phase Shift

Phase shift is the horizontal displacement of the wave from its usual position. It indicates how much the graph is shifted along the x-axis. For a function in the form \(y = a \sin(bx - c)\), the phase shift \(\text{is given by } \frac{c}{b}\).

Analyzing the function \(y = 2 \sin(\pi x - 1)\), we find:\(c = 1\) and \(b = \pi\). Using our formula:

• \(\text{Phase Shift} = \frac{1}{\pi}\)

This represents a shift to the right by a distance of \(\frac{1}{\pi}\). The direction of the shift is determined by examining the expression \(bx - c\). Since we're subtracting \(c\), the shift is in the positive x-direction. This shift affects where the wave starts its cycle along the x-axis, effectively moving the function horizontally.