Problem 40 - Precalculus : Building Concepts and Connections

Question

Graph the given pair of functions on the same set of axes. Are the graphs of \(f\) and \(g\) identical or not? $$f(x)=\cos (x+\pi) ; g(x)=-\cos (x)$$

Step-by-Step Solution

Answer

No, the graphs of \(f(x)=\cos (x+\pi)\) and \(g(x)=-\cos (x)\) are not identical. They differ in both the phase shift and the direction they veer (one is the flipped image of the other).

Step 1: Graphing the Function f(x)

The cosine function is a wave that starts at the point (0,1) and oscillates between -1 and 1. But \(f(x)=\cos (x+\pi)\) has been phase shifted, which means the graph would start from \(\pi\) less than the original cosine function. So the typical start point of (0,1) would now be \((- \pi,1)\) and from there, the wave function will follow.

Step 2: Graphing the function g(x)

The function \(g(x)=-\cos (x)\) is simply the mirror image of the cosine function about the x-axis because we are taking the negative of the function. Therefore the point (0,1) would now be flipped to (0,-1)

Step 3: Comparing the graphs

Upon examining the two graphs, it can be determined if they are identical or not. However, from the transformations, it is clear that their start and end points differ. Meaning their phase shifts are different and amplitudes are also different.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented as \(\cos(x)\). It produces a smooth, repeating wave shape that oscillates between -1 and 1.

This function starts at \((0, 1)\) and completes one full cycle over a period of \(2\pi\). You can think of it as tracing a path on the unit circle, where the x-coordinate corresponds to the cosine value.

• The cosine wave maintains a consistent shape, with peaks at 1 and troughs at -1.
• Its period is the length of one complete wave cycle, which is always \(2\pi\) for the basic cosine function.
• The amplitude, or height of the wave from center line to peak, is 1.

Understanding these basic characteristics is essential for working with more complex transformations like phase shifts and reversals.

Phase Shift

A phase shift in a trigonometric function refers to the horizontal movement of the wave along the x-axis. For the function \(f(x) = \cos(x + \pi)\), the graph is moved horizontally. The \(+\pi\) indicates that every value of \(x\) is shifted \(-\pi\) to the left. This means it starts half a period ahead compared to the standard cosine function.

When graphing, this shift results in a different starting point. For our function:

• The original start point of \((0, 1)\) becomes \((-\pi, 1)\).
• The entire wave pattern moves left by \(\pi\) units, altering the placement of peaks and troughs.
• These shifts play a crucial role in aligning waves and can drastically change their appearances even if the shape remains identical.

Recognizing and applying phase shifts is vital in understanding how trigonometric functions can be manipulated to fit different scenarios.

Function Transformation

Function transformation can occur in various forms, such as translations, reflections, and scaling, impacting how the graph of a function is displayed.

For example, the function \(g(x) = -\cos(x)\) demonstrates a transformation by reflecting the graph over the x-axis. This reflection changes the orientation but not the period or the absolute size of the graph.

• The standard cosine graph's peaks become troughs in the reflected version and vice versa.
• The point \((0, 1)\) in the regular cosine graph now becomes \((0, -1)\).
• This alteration means the wave's topmost points sit at -1 and the bottommost at 1, making it appear upside down.

When analyzing function transformations, one must account for these changes to accurately understand how the basic wave shape has been adjusted.