Problem 6 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\sin x-\frac{1}{2} $$

Step-by-Step Solution

Answer

The graph of function \( f(x) = \sin x - \frac{1}{2} \) has a sinusoid shape of \( \sin x \) graph but translated downwards by 0.5 units. The maximum point is now at (π/2, 1/2) and the minimum point is at (3π/2, -3/2). The sinusoidal axis is at \( y=-\frac{1}{2} \).

Step 1: Establish the sinusoidal axis

In the ordinary sine function (\( \sin x \)), the sinusoidal axis is y=0. Here, the function is translated downwards by 1/2 units. Hence, the sinusoidal axis becomes y=-1/2.

Step 2: Sketch the initial sine curve

Draw the curve of the original function \( \sin x \), which starts at the origin (0,0), heads up to (π/2, 1), then to (π, 0), and then at (3π/2, -1) and finally at (2π, 0). Remember that this is a periodic function, so this pattern repeats itself every 2π.

Step 3: Translate the original graph down by 1/2 units

Now, translate the original sine curve down by 1/2 units to get the curve of the required function. The maximum of the new graph will now be at (π/2, 1/2), and the minimum will be at (3π/2, -3/2).

Step 4: Draw a line to represent the sinusoidal axis

Draw a horizontal line at \( y = -\frac{1}{2} \) to represent the sinusoidal axis. This line should cut through the middle of our translated sine curve.

Step 5: Complete the graph

Ensure the graph has correct periodicity and shape after the translation. The pattern repeats every 2π along the x-axis, similar to the basic sine function. Label values on the y-axis and x-axis to clearly indicate the changes that happened due to translation.

Key Concepts

The Sine Function

The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. Its graph is a smooth, continuous wave that oscillates above and below the x-axis. This is known as the sine wave. The key features of the sine function are:

• Starts at the origin (0,0) when standard.
• Picks at \( \left( \frac{\pi}{2}, 1 \right) \) and troughs at \( \left( \frac{3\pi}{2}, -1 \right) \).
• Repeats every \( 2\pi \), known as one complete cycle or period.

The amplitude of the sine wave, which is the height from the axis to the peak, is 1. This means it stretches from a maximum value of 1 to a minimum of -1. This periodic nature makes the sine function ideal for modeling repetitive patterns or oscillations, such as sound waves or tides.

Understanding Function Translation

Function translation involves shifting the entire graph of a function in various directions without altering its shape. In this context, we translate the sine function.

• Vertical translations move the graph up or down by adding or subtracting a value.
• In the function \( f(x) = \sin x - \frac{1}{2} \), the graph is shifted down by \( \frac{1}{2} \) units.

To perform this, subtract \( \frac{1}{2} \) from every y-coordinate of the original sine wave.
The result is a new sinusoidal axis at \( y = -\frac{1}{2} \), which serves as the baseline for oscillation. This axis now runs right through the center of the wave, making it crucial to understanding and interpreting the translated graph completely.

The Nature of Periodic Functions

Periodic functions like the sine function repeat their values in regular intervals. This repeating nature is what defines periodicity.

• The sine function repeats every \( 2\pi \) radians, known as its period.
• This means after every \( 2\pi \), the graph starts over with the same values as before.

When translation occurs, the period remains unchanged. For the function \( f(x) = \sin x - \frac{1}{2} \), the wave still completes a cycle every \( 2\pi \).
Understanding periodicity is crucial for predicting the behavior of trigonometric functions over different intervals. It ensures the graph maintains its rhythm and structure, no matter how it is transformed or shifted.