Problem 39 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\cos (x+1) $$

Step-by-Step Solution

Answer

The graph of \(f(x)=\cos (x+1)\) is a typical cosine wave shifted one unit to the left.

Step 1: Understanding the basic nature of the cosine function

The fundamental graph of the cosine function, \(y=\cos(x)\), is a wave-like pattern that oscillates between -1 and 1 on the y-axis. It completes a full cycle every \(2\pi\).

Step 2: Identifying the shift in the function

In this particular function \(f(x)=\cos (x+1)\), the +1 inside the cosine function will shift the graph to the left by one unit. This is because the +1 is subtracted from the input x, effectively shifting all points to the left.

Step 3: Drawing the graph

Start by drawing the basic cosine function. Then shift every point one unit left due to \(x+1\). Be sure to label your points and draw your wave to suggest the pattern continues indefinitely in both directions.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions, such as the cosine function, are essential players in mathematics, especially within trigonometry and calculus. These functions relate the angles of a triangle to the ratios of its sides. In the case of the cosine function, it helps to think of it as mapping an angle (or repeated rotation) to a point on the unit circle.

• Cosine Function Basics: The base cosine function, written as \(y = \cos(x)\), creates a wave-like shape. This pattern is important because it repeats itself, or is periodic, in nature. It oscillates between 1 and -1, displaying peaks at multiples of \(2\pi\).
• Full Cycle: The cosine function completes one full wave cycle as \(x\) moves from 0 to \(2\pi\). This is where it begins to repeat its values again.

Understanding this foundation can provide insights as to how modifying the function affects its graph.

Exploring Phase Shift

Phase shift refers to the horizontal movement of the graph of a trigonometric function. It occurs when there is an addition or subtraction within the function's parenthesis, such as in the equation \(f(x) = \cos(x + 1)\).

• Basic Idea: The expression \(x + 1\) means shift. In this equation, adding 1 inside the parenthesis moves the cosine graph to the left by one unit.
• Visualizing: Think of it as sliding the entire wave to the left; each point on the basic \(y = \cos(x)\) graph moves left by one step.
• Implications: This shift changes where the peaks, troughs, and intercepts occur without altering their heights or depths.

By mastering the concept of phase shift, you'll be ready to graph any transformed trigonometric function.

Graph Transformations and How They Work

Graph transformations take the basic form of a trigonometric function and modify its appearance by shifting, stretching, or compressing it. This helps to apply mathematical concepts to a broad range of scientific and engineering problems.

• Types of Transformations: The most common transformations include shifting (up, down, left, right), scaling (stretching or compressing), and reflecting. In the function \(f(x) = \cos(x + 1)\), only a horizontal shift occurs.
• Shift Transformations: Here, introducing a new term like \(x+1\) results in a shift. It does not alter the wave's shape, just its position.
• Application: In visualizing transformations, always start with the original function. Draw it, then apply the transformation step by step to clearly visualize the shift.

Grasping graph transformations allows better comprehension of how changes in equations manifest in their visual representation.