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 = 1/3 sin x

Verified step by step guidance

1. Identify the amplitude of the function y = \(\frac{1}{3}\) \(\sin\) x. The amplitude of a sine function y = A \(\sin\) x is the absolute value of the coefficient A in front of \(\sin\) x.
2. Write down the amplitude as |\(\frac{1}{3}\)|, which represents the maximum distance the sine wave reaches from the horizontal axis (y = 0).
3. To graph the function y = \(\frac{1}{3}\) \(\sin\) x, plot points for values of x between 0 and 2\(\pi\), calculating y by multiplying \(\sin\) x by \(\frac{1}{3}\) at each point.
4. On the same coordinate system, graph y = \(\sin\) x for 0 \(\leq\) x \(\leq\) 2\(\pi\), noting that its amplitude is 1, so it reaches higher and lower peaks compared to y = \(\frac{1}{3}\) \(\sin\) x.
5. Compare the two graphs to observe how the amplitude affects the height of the sine wave, with y = \(\frac{1}{3}\) \(\sin\) x having smaller peaks and troughs than y = \(\sin\) x.

Key Concepts

Amplitude of a Sine Function

The amplitude of a sine function is the absolute value of the coefficient in front of the sine term. It represents the maximum vertical distance from the midline (usually the x-axis) to the peak of the wave. For y = (1/3) sin x, the amplitude is |1/3| = 1/3.

Graphing Sine Functions

Graphing a sine function involves plotting points based on its amplitude, period, phase shift, and vertical shift. The basic sine function y = sin x has an amplitude of 1 and period 2π. Modifying the amplitude scales the graph vertically, affecting the height of peaks and troughs.

Comparing Functions on the Same Coordinate System

Plotting multiple functions on the same axes allows visual comparison of their behaviors. Here, graphing y = (1/3) sin x alongside y = sin x over 0 ≤ x ≤ 2π shows how amplitude changes affect the wave's height while the period and shape remain consistent.