Problem 37 - 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? $$f(x)=\sin (x+\pi) ; g(x)=-\sin x$$

Step-by-Step Solution

Answer

The graphs of \(f\) and \(g\) are identical. Both functions produce a sine wave that completes a full cycle from -pi to pi, albeit inverted relative to each other. However, due to the cyclical nature of the sine function, the graph of \(f(x) = \sin(x + \pi)\) and \(g(x) = -\sin(x)\) is essentially the same.

Step 1: Understand the Functions

First, take a look at the provided functions. Both are based on the sine function, but have variations. \(f(x)=\sin (x+ \pi)\) has a pi phase shift to the left while \(g(x)=-\sin x\) is a reflection of the standard sine wave over the x-axis.

Step 2: Graphing \(f(x)=\sin (x+ \pi)\)

To graph \(f(x)=\sin (x+ \pi)\), plot standard sine wave, shift each point pi units to the left. The sine wave completes a full cycle from -pi to pi, as opposed to the standard [0, 2pi].

Step 3: Graphing \(g(x)=-\sin x\)

To graph \(g(x)=-\sin x\), plot the standard sine wave then reflect it over the x-axis. The positive values on the standard sine wave will become negative on this wave, and vice versa.

Step 4: Compare the Two Graphs

By overlaying or simply comparing the two graphs, it can be seen if they are identical or not. An identical graph will have the exact same shape, position and values as the other.