Problem 171 - Calculus Volume 1

Question

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. $$y=-3 \sin (\pi x+2)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Standard Form

First, we need to identify the standard form of the sine function. The general form is \( y = a \, \sin(bx + c) \). In this case, we have \( y = -3 \, \sin(\pi x + 2) \). Here, \( a = -3 \), \( b = \pi \), and \( c = 2 \).

Step 2: Calculate the Amplitude

The amplitude of a sine function is the absolute value of \( a \). In this function, \( a = -3 \), so the amplitude is \( |a| = 3 \).

Step 3: Determine the Period

The period of a sine function can be found using the formula \( \frac{2\pi}{b} \). Here, \( b = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).

Step 4: Find the Phase Shift

The phase shift of a sine function is given by \( -\frac{c}{b} \). With \( c = 2 \) and \( b = \pi \), the phase shift is \( -\frac{2}{\pi} \). This indicates a shift to the left, because the value is negative.

Key Concepts

Amplitude

The amplitude is an important characteristic of trigonometric functions, especially sine and cosine ones. It tells you how "tall" the wave is—that is, the height from the middle of the wave to its peak or trough. In mathematical terms, the amplitude is the absolute value of the coefficient in front of the sine or cosine function in its standard form.

For example, in a sine function given in the form of \( y = a \sin(bx + c) \), the amplitude is \(|a|\). Let's say you have the function \( y = -3 \sin(\pi x + 2)\). Here, \( a = -3 \). Even though it's negative, the absolute value is used when calculating amplitude. That means the amplitude is \(|-3| = 3\).

A larger amplitude means the function's graph stretches further away from the horizontal axis. The amplitude of \(3\) tells us that the sine wave extends 3 units above and below its midline (usually the x-axis). Knowing the amplitude helps in understanding how much the function value will oscillate as the variable changes.

Period of Sine Function

The period of a sine (or cosine) function refers to the amount of horizontal distance required for the wave to complete one full cycle. For sine functions, the standard period is typically \(2\pi\), but this can vary depending on the value of \(b\) in the standard form \( y = a \sin(bx + c) \).

To find the period of the function, one uses the formula \( \frac{2\pi}{b} \). For the given function \( y = -3 \sin(\pi x + 2)\), \( b = \pi \), resulting in the period \( \frac{2\pi}{\pi} = 2 \).

This calculation means that every 2 units on the x-axis, the sine function completes one full oscillation from its starting point and back again. Changing \(b\) will stretch or compress the waveform along the x-axis, affecting how quickly it repeats its up-and-down cycle. A smaller period indicates a faster repeating wave.

Phase Shift

The phase shift is a measure of how a trigonometric function is horizontally shifted from its usual position. This is determined by the term \(c\) in the standard sine function formula \( y = a \sin(bx + c) \).

To find the phase shift, we use the formula \( -\frac{c}{b} \). Let's apply this to \( y = -3 \sin(\pi x + 2)\), where \( c = 2 \) and \( b = \pi \). The phase shift is \( -\frac{2}{\pi} \). The negative sign indicates a leftward shift on the x-axis.

The phase shift tells us how much the function is moved from where it would normally start. It can significantly affect the function's graph, changing the points where the wave first reaches its peak, crosses the midline, or hits its trough. Solving for the phase shift helps in accurately plotting and understanding the wave's behavior in context with other sinusoidal functions.