Problem 1 - Algebra and Trigonometry

Question

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=4 \sin x$$

Step-by-Step Solution

Answer

The amplitude of the function y=4sinx is 4. When graphed, y=sinx and y=4sinx will have similar shapes. However, the function y=4sinx will reach a maximum height of 4 and a minimum height of -4 while y=sinx will only reach a maximum and minimum of 1 and -1 respectively.

Step 1: Determine the amplitude

The amplitude is the coefficient of the sine function, which is the number multiplied by the sine of x. So in this equation y=4sinx, the amplitude is 4.

Step 2: Plotting the functions

In a rectangular coordinate system, draw the y=sinx graph first. Remember that sinx has an amplitude of one which means the peak and trough are at y=1 and y=-1. It completes one cycle in a range of 2π. Now, draw y=4sinx. The amplitude here is 4, so the highest and lowest points of the wave are at y=4 and y=-4. Additionally, y=4sinx traverses through one full sinusoidal period within 2π same as sinx.

Key Concepts

Amplitude of sine functions

The amplitude of a sine function is a fundamental concept in trigonometry. It defines the height of the peaks and the depth of the troughs from the centerline or the x-axis, giving a clear idea of the function's "stretch." For the sine function, the amplitude is directly given by the coefficient placed before the sine term. For example, in the function \( y = 4 \sin x \), the number 4 is the amplitude. This means that the wave will reach its maximum at \( y = 4 \) and its minimum at \( y = -4 \).

• The sine function generally has a form \( y = a \sin x \), where \( a \) is the amplitude.
• If \( a \,<0 \), the graph is vertically flipped but the amplitude is still \( |a| \).
• The absolute value of \( a \), \( |a| \), determines the amplitude.

Consequently, understanding the amplitude is crucial for comprehending the magnitude of oscillation within a sine function and how it modifies the standard sine curve.

Graphing sine functions

Graphing sine functions provide a visual representation of their behavior across a specified interval, typically \( 0 \leq x \leq 2\pi \). The graph of a sine function generally showcases its periodic nature and amplitude.Begin by plotting the standard sine function \( y = \sin x \), which has:

• An amplitude of 1, indicating the wave peaks at 1 and troughs at -1.
• One complete cycle over \( 2\pi \), meaning the distance between identical points on the graph, like from one peak to the next, is \( 2\pi \).

Now, to graph \( y = 4 \sin x \):

• Start at the origin \((0,0)\), similar to \( y = \sin x \).


• Note the amplitude change; peaks and troughs are now at 4 and -4, significantly stretching the graph vertically.


• Complete one sinusoidal cycle between 0 to \( 2\pi \), similar to \( y = \sin x \), by plotting key points at \( 0, \pi/2, \pi, 3\pi/2, \text{and} \, 2\pi \).

Graphing these functions allows you to better see the effect of the amplitude on the shape and range of the waveform.

Periodicity of trigonometric functions

The periodicity of trigonometric functions refers to the recurring nature of their graphs at regular intervals. For sine functions, this concept is quite visible in their consistent wave patterns over a given range, typically \(0\) to \(2\pi \).The period of a standard \( y = \sin x \) function is \(2\pi \), which means every \(2\pi \) units on the x-axis, the function begins to repeat its cycle.
- For the function \( y = 4 \sin x \), even though the amplitude is different, it still retains this fundamental period of \(2\pi \).- This means the wave we plot will look like one "hump" or "valley" between 0 and \(2\pi\) – repeating this structure again immediately after.- The periodicity ensures that for any sine function, no matter how stretched or compressed vertically by the amplitude, the horizontal length needed for a full cycle in \(y = \sin x\) is still \(2\pi\).Understanding periodicity in trigonometric functions is important for predicting their values over the entire function's graph — a useful property both in math and practical applications like electronics or signal processing.