Problem 34 - Algebra 2

Question

Write an equation for each translation. \(y=\cos x, 1.5\) units to the right

Step-by-Step Solution

Answer

The resulting equation after translating the cosine function 1.5 units to the right is \(y = \cos (x-1.5)\).

Step 1: Identify the Original Function

The original function provided in the exercise is \(y = \cos x\). This function has a principal period \(2\pi\), and it is repeated in intervals of \(2\pi\).

Step 2: Recognize the Translation

The problem has stated that the cosine function will be translated 1.5 units to the right. Shifting functions horizontally may involve adjusting the function's argument. A shift to the right indicates subtraction, while a move to the left implies addition.

Step 3: Apply the Translation

Taking into account a right shift by 1.5 units, we need to subtract 1.5 from the function's argument \(x\). Thus, the translated function becomes \(y = \cos (x-1.5)\).

Key Concepts

Horizontal Shift

In trigonometric functions, a horizontal shift involves moving the graph of the function left or right on the x-axis. This shift is achieved by altering the argument of the function, which means we adjust the variable inside the cosine, sine, or other trigonometric functions. For a shift to the right, we subtract the desired shift amount from the variable. To shift a function to the left, we add the desired shift value to the variable.

When tackling problems involving horizontal shifts, remember:

• Subtract from the variable to shift right (e.g., \(y = \cos(x-1.5)\)).
• Add to the variable to shift left (e.g., \(y = \cos(x+1.5)\)).

Horizontal shifts do not affect the amplitude or period of the trigonometric function, only the starting point of each cycle.

Cosine Function

The cosine function, written as \(y = \cos x\), is a fundamental part of trigonometry. It represents one of the basic trigonometric functions, which is periodic in nature. The cosine function oscillates between -1 and 1, producing a wave-like pattern. Its period is \(2\pi\), meaning the pattern repeats every \(2\pi\) units along the x-axis.

Some characteristics of the cosine function include:

• Amplitude: The maximum or minimum value, which is 1 for the cosine function.
• Period: The length of one full cycle, \(2\pi\).
• Phase Shift: The horizontal shift along the x-axis.
• Midline: The equilibrium position, located at \(y=0\), the horizontal line around which the function oscillates.

Understanding the cosine function helps in analyzing periodic phenomena, such as sound waves or alternating current electric signals.

Trigonometric Functions

Trigonometric functions like the sine, cosine, and tangent are fundamental to the study of mathematics, particularly in geometry and engineering. These functions are defined based on the ratios of the sides of right-angled triangles, and they repeat values in cycles due to their periodic nature.

Important aspects of trigonometric functions include:

• Amplitude: Reflects how far the function's value can reach above or below its midline.
• Period: Determines how long it takes for the function to complete one full cycle.
• Phase Shift: Adjusts the horizontal position of the graph. Involves moving the function left or right along the x-axis.
• Other transformations: Vertical shifts and reflections can also modify these functions.

These functions are not only confined to triangles but have applications in various fields, such as physics, where they model wave and oscillation behaviors. Being tuned into how trigonometric functions translate, rotate, and stretch can unveil powerful insights into natural and man-made phenomena.