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Chapter 4: Problem 98

Graph \(f\) and \(g\) in the same coordinate plane. Include two full periods. Make a conjecture about the functions. $$f(x)=\sin x, \quad g(x)=-\cos \left(x+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The graphs of the functions \(f(x)=\sin x\) and \(g(x)=-\cos (x+\frac{\pi}{2})\) have the same shape but are reflections of each other across the x-axis. Therefore, the conjecture is that these two functions are equivalent but negative reflections of each other in their graphs.

Step by step solution

01

Determine the Characteristics of the Function \(f(x)=\sin x\)

The function \(f(x)=\sin x\) is a sinusoidal function with an amplitude of 1, a period of \(2\pi\), no horizontal shift, and no vertical shift. It starts at the origin (0,0) and goes up to (1,0), then down to (-1,0).
02

Determine the Characteristics of the Function \(g(x)=-\cos (x+ \frac{\pi}{2})\)

The function \(g(x)=-\cos (x+\frac{\pi}{2})\) is a cosine function that is vertically reflected over the x-axis, and horizontally shifted to the left by \(\frac{\pi}{2}\) units. Its amplitude is also 1 and its period is \(2\pi\). It starts at the origin (0,0), goes down to (1,0) and then up to (-1,0).
03

Graphing the Functions

Plot the graphs of the two functions in the same coordinate plane using their period. The x-axis is divided into intervals of \(\pi/2\) for reflecting the period. The y-axis represents the values of \(f(x)\) and \(g(x)\). Plotting the points accurately can better show two full periods of the functions.
04

Make a Conjecture

By observing the two full periods of the graphs, it can be seen that \(f(x)) = g(x)\). This is because both functions have the same period, same amplitude, and start at the origin (0,0), the only difference is the direction in which they go from the origin. Hence, it can be conjectured that the functions are equivalent but negative reflections of each other across the x-axis.

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