Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{array}{l} f(x)=\sin x \\ g(x)=\sin (x-\pi) \end{array}$$

Step-by-Step Solution

Answer

Both functions have the same amplitude of 1 and period of \(2\pi\). However, \(g(x)\) is shifted \(\pi\) units to the right compared to \(f(x)\).

Step 1: Analyze the Amplitude

The amplitude of a sine function is the half distance from its minimum to maximum. In this case, both \(f(x)=\sin x\) and \(g(x)=\sin (x-\pi)\) have the same amplitude of 1 since there are no coefficients in front of the sine function.

Step 2: Analyze the Period

The period of a sine function is the distance over which it repeats its pattern. For standard sine function, the period is \(2\pi\). Here, both \(f(x)=\sin x\) and \(g(x)=\sin (x-\pi)\) share the same period of \(2\pi\).

Step 3: Analyze the Shift

The shift of a sinusoidal function refers to changes in their phase which denotes horizontal movements. Function \(f(x)=\sin x\) has no phase shift as it is a standard sine function. However for function \(g(x)=\sin (x-\pi)\), it has a phase shift equal to \(\pi\), meaning it is shifted \(\pi\) units to the right compared to \(f(x)\).

Key Concepts

Amplitude of Sine Function

When observing sine functions, one key feature to consider is the amplitude, which reflects the wave's height. In layman's terms, imagine a wave at sea; the amplitude would be the measurement from the calm sea level to the wave's peak - it determines how 'tall' the wave is. Mathematically, it is defined as the distance from the function's average value (usually the midpoint between its peak and trough) to its maximum or minimum.

For the function f(x) = sin x, the amplitude is straightforward because the maximum height reached by the sine graph is 1 and the minimum is -1. There are no modifiers changing these values. The same goes for g(x) = sin (x - \(\pi\)). Both of these functions have an amplitude of 1, demonstrating that their peak-to-peak variation will be identical. When analyzing textbook exercises, understanding and identifying the amplitude can quickly give you insights into the graphical representation of a sine function.

Period of Sine Function

The period of a sine function is the length of one full cycle of the wave and represents how 'wide' a single oscillation is. It's similar to the distance between two successive crests or troughs of a wave. The standard sine function, f(x) = sin x, completes one full cycle of its up-and-down motion over an interval of 2\(\pi\) radians.

Both f(x) and g(x) = sin (x - \(\pi\)) in our exercise share this period of 2\(\pi\). The significance of the period comes into play when the input variable x is scaled or stretched. If, for instance, we had a coefficient in front of x, such as in sin(2x), the period would be halved to \(\pi\) because the wave would complete its cycle more quickly.

Phase Shift in Trigonometry

A phase shift in trigonometry is akin to sliding a function left or right along the x-axis without altering its shape. It's like relocating a slide in a playground; the fun it provides remains the same, but its position changes. For sine functions, a phase shift is indicative of where the wave starts.

In the comparison between f(x) = sin x and g(x) = sin (x - \(\pi\)), we see that while f(x) starts at the wave's natural origin, g(x) begins \(\pi\) units to the right. This means each point on the graph of g(x) is shifted \(\pi\) units horizontally to the right compared to f(x). It's crucial to accurately determine the phase shift to predict where on the x-axis the waveform will be located.

Sinusoidal Function Transformations

Sinusoidal function transformations encompass a variety of changes – including shifts, stretches, and reflections – that can be applied to the base sine function. Think of it as choosing different settings for your favorite song; you might increase the volume (amplitude), speed it up (period), or even play it in reverse (reflection). With transformations, the sine wave can be flipped vertically, compressed or expanded horizontally, and moved left or right along the x-axis.

In the functions we are looking at, the transformation comes in the form of the aforementioned phase shift. Specifically, g(x) is the result of applying a horizontal shift to f(x). These transformations enable us to model real-world phenomena using sine waves accurately. Understanding how to apply and interpret these transformations is key to mastering trigonometry and its applications.