Problem 14 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\sin \left(x-\frac{\pi}{4}\right)-1 $$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\sin \left(x-\frac{\pi}{4}\right)-1\) is a sine wave which is shifted \(\frac{\pi}{4}\) units to the right, and 1 unit down.

Step 1: Understand the function

The function \(f(x)=\sin \left(x-\frac{\pi}{4}\right)-1\) is a simple transformation of the sine function. It involves a horizontal shift to the right by \(\frac{\pi}{4}\), and a vertical shift down by 1 unit.

Step 2: Plot the original sine function

Start by plotting the graph of the standard sine function, \( y = \sin(x) \). The sine function has a period of \(2\pi\), so plot the graph accordingly over a range that includes at least one complete period.

Step 3: Horizontal Shift

The term \(x-\frac{\pi}{4}\) shifts the graph of the sine function \(\frac{\pi}{4}\) units to the right. Adjust the graph accordingly, moving each point \(\frac{\pi}{4}\) units to the right.

Step 4: Vertical Shift

The term -1 in the function shifts the graph down by 1 unit. Shift each point on the graph down by one.

Step 5: Final Graph

After applying these transformations, the final graph represents the function \(f(x)=\sin \left(x-\frac{\pi}{4}\right)-1\).

Key Concepts

Understanding the Sine Function

The sine function, denoted as \( \sin(x) \), is one of the foundational trigonometric functions used to describe repeating patterns, like waves. This function produces a graph resembling a wave that oscillates smoothly between -1 and 1. The standard form of the sine function has some characteristics that make it unique:

• The curve starts at the origin, \( (0,0) \), when plotted on a graph.
• Its maximum value is 1, and its minimum value is -1.
• It is a periodic function, repeating its shape every \( 2\pi \) radians.

Sine functions are largely used to model periodic phenomena such as sound waves, light waves, and alternating current in electrical circuits. When graphing the standard sine function, remember these key points to help you identify and transform the graph accurately.

Graph Transformations

Transformations alter the way a function looks on a graph without changing its core characteristics. With the function \( f(x)=\sin \left(x-\frac{\pi}{4}\right)-1 \), two main transformations have been applied:

• Horizontal Shift: The expression \( x-\frac{\pi}{4} \) indicates a horizontal shift. Here, the graph of \( \sin(x) \) is moved to the right by \( \frac{\pi}{4} \) units. To apply this shift, move each point of the original graph rightwards by \( \frac{\pi}{4} \) along the x-axis.
• Vertical Shift: The "-1" at the end of the function implies a downward vertical shift. Every point on the graph of the modified sine function is translated 1 unit downward.

Graph transformations like these don't modify the function's period or waveform but change the starting point and baseline of the cycle, allowing the sine function to model different situations and datasets by repositioning its key features.

Exploring Periodic Functions

Periodic functions are mathematical functions that repeat their values at regular intervals, known as periods. The sine function exhibits this periodicity with its repeating wave pattern. Understanding periodic functions is crucial because:

• Repetitive Behavior: They model phenomena that have consistent cycles, such as tides, seasons, or alternating current.
• Period Identification: The period is the distance over which the function completes one full cycle. For sine, this period is \( 2\pi \), meaning after every \( 2\pi \) interval, the wave pattern starts anew.
• Predictability: Given that they repeat identically, periodic functions allow for predictions over extended periods, making them invaluable in science and engineering.

By recognizing the periodic nature of functions like the sine, one can effectively anticipate and analyze patterns in real-world data, extending these insights into various fields from acoustics to meteorology.