Problem 21 - Thomas Calculus

Question

Graph the functions in Exercises \(13-22 .\) What is the period of each function? $$\sin \left(x-\frac{\pi}{4}\right)+1$$

Step-by-Step Solution

Answer

The period of the function is \(2\pi\).

Step 1: Identify the Base Function

The given function is \( \sin \left(x-\frac{\pi}{4}\right)+1 \). The base function here is the sine function \( \sin(x) \). We will analyze how this base function is transformed.

Step 2: Determine Horizontal Shift

The expression inside the sine function is \( x - \frac{\pi}{4} \). This indicates a horizontal shift to the right by \( \frac{\pi}{4} \). The graph of the sine function will start \( \frac{\pi}{4} \) units to the right.

Step 3: Determine Vertical Shift

The entire function is \( \sin \left(x-\frac{\pi}{4}\right)+1 \). The \(+1\) denotes a vertical shift upwards by 1 unit. Each point on the sine graph is moved 1 unit up.

Step 4: Identify Period of Sine Function

The period of the basic sine function \( \sin(x) \) is \( 2\pi \). This period is not altered by horizontal or vertical shifts, so it remains \( 2\pi \) for this function.

Step 5: Conclusion on Period

Even after incorporating shifts (horizontal or vertical), the period of \( \sin \left(x-\frac{\pi}{4}\right)+1 \) remains \( 2\pi \). Therefore, the function completes one full cycle every \( 2\pi \) units.

Key Concepts

Sine Function

The sine function, represented as \(\sin(x)\), is a fundamental trigonometric function that describes a smooth, periodic wave. The graph of the basic sine function is a continuous wave that oscillates between -1 and 1, moving above and below the x-axis. Each complete wave cycle of \(\sin(x)\) repeats over an interval known as the "period." This inherent property makes sine waves crucial in modeling repetitive phenomena such as sound or light waves.

• The sine wave starts at the origin \((0,0)\) and rises to a maximum point of 1 at \(\frac{\pi}{2}\).
• It returns to zero at \(\pi\), dips to a minimum of -1 at \(\frac{3\pi}{2}\), and completes the cycle back at zero at \(2\pi\).

Understanding the base form of the sine function is essential for analyzing any transformations, such as shifts or stretches.

Period of Function

The period of a function, especially in trigonometry, refers to the length of one complete cycle of the wave before it repeats. For the basic sine function \(\sin(x)\), the period is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the sine wave starts a new cycle. The period is crucial when graphing as it dictates the spacing of the repeating pattern.

• For \(\sin(x)\), alterations in the equation like \(\sin(kx)\) can affect the period, where the period becomes \(\frac{2\pi}{k}\).
• However, shifts in the sine function, such as horizontal or vertical shifts (like in the given function), do not change the period; it remains \(2\pi\).

Recognizing that the period remains constant despite transformations simplifies graphing and understanding the sine function in various situations.

Function Transformations

Function transformations are modifications that alter the appearance of the graph without changing the fundamental nature of the function. When dealing with transformations of the sine function, the main changes include vertical and horizontal shifts, as well as reflections and dilations.

• Vertical Shifts: Adding or subtracting a constant (e.g., \(+1\) in \(\sin(x) + 1\)) moves the entire graph up or down accordingly.
• Horizontal Shifts: Adjustments inside the function's argument, such as \(x - \frac{\pi}{4}\), shift the graph left or right. The given function shifts the sine wave right by \(\frac{\pi}{4}\) units.
• Amplitude Changes: Altering the coefficient of \(\sin\) (unseen in this problem) affects the wave’s height.

Understanding these transformations helps in predicting and sketching the new graph quickly. Every component in the equation changes something specific and learning to recognize these patterns allows for an intuitive grasp of function behavior.