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Chapter 4: Problem 22

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitude, period, and shifts. $$\begin{aligned} &f(x)=\sin 3 x\\\ &g(x)=\sin (-3 x) \end{aligned}$$

Short Answer

Expert verified
Both functions have the same amplitude (1) and period (\( \frac{2\pi}{3} \)). The function \(g(x) = \sin(-3x)\) is a reflection of \(f(x) = \sin(3x)\) over the y-axis, meaning that it's the same shape but flipped horizontally.

Step by step solution

01

Analyzing the Amplitudes

The amplitude of a sine function is the absolute value of the coefficient of the sine, that is the number in front of 'sin' in the equation. For both \(f(x) = \sin(3x)\) and \(g(x) = \sin(-3x)\), the amplitude is 1, as there is no number multiplied by 'sin'.
02

Analyzing the Periods

The period of a standard sine function is \(2\pi\) and the coefficient of \(x\) inside the function affects the period of function as follows: if the coefficient of \(x\) is \(n\), the period become \( \frac{2\pi}{\left| n \right|} \). For both \( f(x) = \sin(3x) \) and \( g(x) = \sin(-3x) \), the coefficient of \(x\) in the bracket are \(3\) and \(-3\) respectively, so the period of both functions is \( \frac{2\pi}{3} \).
03

Analyzing the Shifts

The sine function \(f(x) = \sin(3x)\) has neither horizontal nor vertical shift as there is no addition or subtraction either outside the function or within the argument of sine function. The function \(g(x) = \sin(-3x)\), however, will be reflected across the y-axis compared to \(f(x)\), because the negative sign inside the function causes a reflection.
04

Comparing the Two Functions

Both functions have an amplitude of 1 and a period of \( \frac{2\pi}{3} \), but the function \(g(x) = \sin(-3x)\) is a reflection of \(f(x) = \sin(3x)\) over the y-axis.

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