Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\sin \left(x+\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

Amplitude: 1; Period: \(2\pi\); Phase shift: \(-\frac{\pi}{4}\) (left).

Step 1: Identify the Amplitude

For the function \(y = a \sin(bx + c) + d\), the amplitude is \(|a|\). In this exercise, the sine function is \(y = \sin\left(x + \frac{\pi}{4}\right)\), where \(a = 1\). Thus, the amplitude is \(|1| = 1\).

Step 2: Determine the Period

The period of a sine function \(y = a \sin(bx + c) + d\) is given by \(\frac{2\pi}{|b|}\). Here, \(b = 1\), resulting in a period of \(\frac{2\pi}{1} = 2\pi\).

Step 3: Find the Phase Shift

The phase shift is calculated as \(-\frac{c}{b}\). In this case, \(c = \frac{\pi}{4}\) and \(b = 1\), leading to a phase shift of \(-\frac{\pi}{4}\). That means the graph is shifted to the left by \(\frac{\pi}{4}\).

Step 4: Sketch the Graph

To sketch the graph, start with the regular sine wave. Then apply the phase shift by moving every point on the sine wave to the left by \(\frac{\pi}{4}\). The amplitude (1) and period (\(2\pi\)) remain unchanged. The resulting graph shows a standard sine wave beginning at \(-\frac{\pi}{4}\).

Key Concepts

Amplitude

When we talk about the amplitude of a trigonometric function, we're discussing the height of its peaks and the depth of its valleys. In simpler terms, it tells us how tall or short the wave is. For a function of the form \(y = a \sin(bx + c) + d\), the amplitude is determined by the coefficient \(|a|\). The absolute value of \(a\) ensures we always have a positive measure for the amplitude since height cannot be negative.

• The amplitude directly influences the maximum and minimum values that the sine function will reach, oscillating between \(-a\) and \(a\).
• In the given exercise, since \(a = 1\), the amplitude is simply \(|1| = 1\).

This means that the function will reach a maximum height of 1 and a minimum of -1. The standard sine function doesn't stretch or shrink vertically, maintaining these bounds. Understanding amplitude is fundamental because it affects the vertical characteristics of the wave, but does not influence the horizontal placement or spacing of the waves.

Period

The period of a trigonometric function like sine or cosine describes the length of one complete cycle of the wave before it starts repeating. It's like the wavelength in physics. For sine functions described by \(y = a \sin(bx + c) + d\), the period is found using the formula \(\frac{2\pi}{|b|}\).

• The variable \(b\) affects how quickly the sine function oscillates.
• A larger \(|b|\) results in a shorter period, meaning more cycles occur over a fixed interval.
• In contrast, a smaller \(|b|\) stretches out each cycle over a longer interval.

For our function \(y = \sin(x + \frac{\pi}{4})\), \(b = 1\), leading to a period of \(2\pi\). This indicates that each cycle of the sine wave takes \(2\pi\) units on the \(x\)-axis before it begins to repeat. While the amplitude describes vertical behavior, the period helps us understand how condensed or spread out each wave cycle is along the \(x\)-axis.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric wave along the \(x\)-axis. It's like sliding a graph left or right without changing its shape. For the function \(y = a \sin(bx + c) + d\), the phase shift is calculated by \(-\frac{c}{b}\).

• A positive phase shift moves the graph to the left, while a negative one moves it to the right.
• The expression \(-\frac{c}{b}\) tells us how many units to shift the wave.

In the exercise's function \(y = \sin(x + \frac{\pi}{4})\), we have \(c = \frac{\pi}{4}\) and \(b = 1\), leading to a phase shift of \(-\frac{\pi}{4}\). This means the entire sine wave is shifted \(\frac{\pi}{4}\) units to the left.
Understanding the phase shift is crucial for graphing these functions because it aids in accurately capturing where each wave cycle begins and ends on the \(x\)-axis, giving insight into how the wave is positioned relative to the origin.