Problem 6 - Precalculus: Functions and Graphs

Question

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: \(-\frac{\pi}{4}\).

Step 1: Understand the General Form of Sine Function

The general form of a sine function is given by \( y = a \sin(bx + c) + d \). Here, \( a \) is the amplitude, \( b \) affects the period, \( c \) influences the phase shift, and \( d \) causes vertical shifts. In our function \( y = \sin \left(x + \frac{\pi}{4}\right) \), we have \( a = 1 \), \( b = 1 \), \( c = \frac{\pi}{4} \), and \( d = 0 \).

Step 2: Determine the Amplitude

The amplitude of a sine function is given by the absolute value of \( a \). Since \( a = 1 \), the amplitude is \( |1| = 1 \).

Step 3: Calculate the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \). In this function, \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Find the Phase Shift

The phase shift is found by solving \( bx + c = 0 \) for \( x \). For the function \( y = \sin(x + \frac{\pi}{4}) \), set \( x + \frac{\pi}{4} = 0 \). Solving for \( x \), we get \( x = -\frac{\pi}{4} \). This means the graph is shifted \( \frac{\pi}{4} \) units to the left.

Step 5: Sketch the Graph

To sketch the graph, know that the basic sine wave starts at the origin. With an amplitude of 1, it stretches between \( y = 1 \) and \( y = -1 \). The period \( 2\pi \) means one full wave cycles every \( 2\pi \) units on the x-axis, but due to the phase shift \( \frac{\pi}{4} \), the wave starts \( \frac{\pi}{4} \) units left of the usual start.

Key Concepts

Amplitude

Amplitude is a crucial concept when it comes to understanding trigonometric functions like sine and cosine. It essentially tells us how tall the waves of a function are. In the context of our sine function, the amplitude is represented by the coefficient in front of the sine term.

• In the equation \(y = \sin \left(x + \frac{\pi}{4}\right)\), the coefficient is 1.
• The amplitude is calculated as the absolute value of this coefficient, meaning the amplitude is \(|1| = 1\).

This tells us the sine wave will oscillate between 1 and -1 on the y-axis. Amplitude is always a positive value, since it represents the maximum wave height from the center line of the graph (y = 0).

Amplitude affects the visual representation of the graph, making waves appear taller or shorter.

Understanding the amplitude helps us determine the range of the function, which plays a big role in sketching the graph accurately.

Period

The period of a trigonometric function is defined as the length of one complete cycle of the wave. For the standard sine and cosine functions, this cycle typically takes place over an interval of \(2\pi\) units.In our specific function \(y = \sin \left(x + \frac{\pi}{4}\right)\), the period is determined by the coefficient of the variable x inside the sine function, identified in the general form as 'b'.

• Here, \(b = 1\) in the function \(y = \sin(x + \frac{\pi}{4})\).
• The formula to find the period of a sine function is \(\frac{2\pi}{b}\).

If we apply this to our equation:* The period is \((2\pi) / 1 = 2\pi\), which implies the function completes one cycle every \(2\pi\) units on the x-axis.It's important to grasp the concept of the period because it informs us about the frequency of the wave's repetition. This knowledge allows for accurate graphing over desired intervals, and it gives us insights about how tightly or widely the waves are spread along the x-axis.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric graph in relation to its usual position. For sine and cosine functions, the standard graph starts at the origin (0,0). However, when a phase shift is present, the graph is moved left or right.To find the phase shift of a sine wave, we look at the expression inside the sine function. The general formula used is to solve \(bx + c = 0\) for x.In our example, \(y = \sin(x + \frac{\pi}{4})\), the elements are:

• \(b = 1\) and \(c = \frac{\pi}{4}\).

Solving \(x + \frac{\pi}{4} = 0\), we find:

• \(x = -\frac{\pi}{4}\).
• This implies a phase shift of \(\frac{\pi}{4}\) units to the left.

Understanding phase shift is crucial in accurately sketching a sine curve as it affects where the wave begins its first cycle. It shifts the pattern along the x-axis, impacting how the graph aligns with other comparative graph locations. Recognizing this helps ensure accurate graphs and facilitates understanding of how changes in the equation shift the graph's entire pattern.