Question

Graph the function. \(g(x)=\cos \left(x-\frac{\pi}{2}\right)\)

Short answer

The graph of \(\cos \left(x-\frac{\pi}{2}\right)\) is a cosine wave that has been shifted to the right by \(\frac{\pi}{2}\) units.

Step-by-step solution

Understand the parent function

First, know that the parent function of \(\cos \left(x-\frac{\pi}{2}\right)\) is \(\cos(x)\). The graph of \(\cos(x)\) is a wave that oscillates between -1 and 1, starting from a maximum point (1) at \(x=0\).

Identify the phase shift

The term in the parenthesis, \(\left(x-\frac{\pi}{2}\right)\), indicates the phase shift of the function. In this case, the function \(\cos(x)\) has been shifted to the right by \(\frac{\pi}{2}\) units.

Plotting the original and the shifted graph

Start by plotting the parent function, \(\cos(x)\). Then, shift every point on the graph to the right by \(\frac{\pi}{2}\) units to get the graph of \(\cos \left(x-\frac{\pi}{2}\right)\). The shape of the function remains the same (a cosine wave), only its location changes.

Key concepts

Cosine Function

When learning about trigonometry, becoming familiar with the cosine function is essential. The function itself represents a fundamental relationship within a right-angled triangle, relating the ratio of the length of the adjacent side to the length of the hypotenuse.

The equation for the basic cosine function is written as \( y = \cos(x) \). The graph of the cosine function is a smooth, continuous wave that flows horizontally along an axis. This wave is known as a 'cosine wave', and has several distinct characteristics:

• It has a period of \( 2\pi \), meaning the wave repeats itself every \( 2\pi \) units along the x-axis.
• The amplitude, or height, of the wave is 1, signifying the maximum value the cosine function can attain.
• The wave oscillates between a maximum value of 1 and minimum value of -1.
• It starts at its maximum at \( x = 0 \) and moves through one complete cycle as \( x \) increases or decreases by \( 2\pi \) units.

Understanding the basic shape and properties of this function aids in comprehending more complex transformations such as phase shifts and vertical/horizontal translations.

Phase Shift

In the realm of trigonometric functions, the phase shift is an alteration that slides the function horizontally along the x-axis. It's a type of transformation changing where the cycle of the trigonometric function begins. For the cosine function, a phase shift doesn't affect the shape or amplitude of the wave; it only moves its starting point.

The phase shift can be detected in the function's equation by looking at the horizontal shift value added or subtracted from the variable \( x \). For example, in the function \( g(x) = \cos \left(x - \frac{\pi}{2}\right) \), the \( -\frac{\pi}{2} \) term signifies a phase shift. Specifically, it translates to shifting the cosine wave to the right by \( \frac{\pi}{2} \) units. This is a critical concept to grasp as it enables you to predict and sketch how trigonometric functions will look after their phase has been altered.

Parent Function

The concept of a parent function is like the blueprint for family traits in mathematics; it is the simplest form of a set of functions that share characteristics. For trigonometric functions, the parent functions are the most basic forms without any transformations such as stretches, compressions, shifts, or reflections.

For the cosine function, the parent function is \( y = \cos(x) \). This represents the pure, undisturbed wave pattern of a cosine curve serving as the foundation for understanding how various changes will affect the graph. When you encounter functions like \( g(x) = \cos \left(x - \frac{\pi}{2}\right) \), identifying that \( \cos(x) \) is the parent function allows you to apply transformations systematically. This knowledge is fundamental when learning to graph the trigonometric functions, as it provides the starting point from which all other transformations are applied.