In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = -3 sin x

Verified step by step guidance

1. Identify the general form of the sine function, which is \( y = a \sin(bx + c) + d \). In this case, the function is \( y = -3 \sin x \).
2. Recognize that the amplitude of a sine function is given by the absolute value of the coefficient \( a \). Here, \( a = -3 \).
3. Calculate the amplitude by taking the absolute value of \( a \), which is \( |a| = |-3| \).
4. Graph the function \( y = -3 \sin x \) and \( y = \sin x \) on the same coordinate system for the interval \( 0 \leq x \leq 2\pi \).
5. Observe that the graph of \( y = -3 \sin x \) has the same period as \( y = \sin x \), but its peaks and troughs are 3 times further from the x-axis, reflecting the amplitude.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave or periodic function reaches from its central axis. In the context of sine functions, it is the absolute value of the coefficient in front of the sine term. For the function y = -3 sin x, the amplitude is 3, indicating that the graph oscillates 3 units above and below the horizontal axis.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the values of the function over a specified interval. For sine functions, the graph typically oscillates between its maximum and minimum values, determined by the amplitude and vertical shift. In this case, y = -3 sin x will have its peaks at -3 and troughs at 0, creating a wave pattern that is inverted due to the negative sign.

Period of Sine Function

The period of a sine function is the length of one complete cycle of the wave. For the standard sine function, the period is 2π, meaning it repeats every 2π units along the x-axis. In the case of y = -3 sin x, the period remains 2π, allowing for consistent comparison with the standard sine function y = sin x over the interval 0 ≤ x ≤ 2π.