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 = 4 sin x

Verified step by step guidance

1. Identify the general form of the sine function, which is \(y = A \sin x\), where \(A\) represents the amplitude.
2. Recall that the amplitude of a sine function is the absolute value of the coefficient in front of \(\sin x\), so amplitude \(= |A|\).
3. In the given function \(y = 4 \sin x\), the coefficient \(A\) is 4, so the amplitude is \(|4|\).
4. To graph the function \(y = 4 \sin x\) along with \(y = \sin x\) on the same coordinate system for \(0 \leq x \leq 2\pi\), plot points for both functions at key values of \(x\) such as \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\).
5. Note that \(y = 4 \sin x\) will have peaks at \(4\) and troughs at \(-4\), while \(y = \sin x\) has peaks at \(1\) and troughs at \(-1\), so the graph of \(y = 4 \sin x\) is a vertically stretched version of \(y = \sin x\) by a factor of 4.

Key Concepts

Amplitude of a Sine Function

The amplitude of a sine function y = a sin x is the absolute value of the coefficient a. It represents the maximum vertical distance from the midline (usually the x-axis) to the peak of the wave. For y = 4 sin x, the amplitude is 4, meaning the graph oscillates between -4 and 4.

Graphing Sine Functions

Graphing a sine function involves plotting points based on its amplitude, period, and phase shift. The basic sine curve y = sin x oscillates between -1 and 1 with a period of 2π. For y = 4 sin x, the shape is similar but scaled vertically by a factor of 4, stretching the wave.

Comparing Functions on the Same Coordinate System

Plotting y = 4 sin x and y = sin x together helps visualize differences in amplitude. Both share the same period and phase but differ in vertical stretch. This comparison highlights how amplitude affects the height of the sine wave without changing its frequency or phase.