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Chapter 4: Problem 58

Use your graphing calculator to graph each family of functions for \(-2 \pi \leq x \leq 2 \pi\) together on a single coordinate system. (Make sure your calculator is set to radian mode.) What effect does the value of \(A\) have on the graph? \(y=A \sin x\) for \(A=1, \frac{1}{2}, \frac{1}{3}\)

Short Answer

Expert verified
The value of \( A \) affects the amplitude of the sine graph. Larger \( A \) values increase the graph's height from the x-axis.

Step by step solution

01

Set the Calculator Mode

Ensure your graphing calculator is set to radian mode since the functions involve trigonometric calculations in terms of . Check your calculator settings to switch from degree mode to radian mode if necessary.
02

Enter the Functions

Input the given functions into the graphing calculator. These are: 1. \( y = 1 \cdot \sin x \)2. \( y = \frac{1}{2} \cdot \sin x \)3. \( y = \frac{1}{3} \cdot \sin x \). Make sure each function is entered separately so they can all be plotted on the same graph.
03

Adjust the Window Settings

Set the domain for the graph to range from \(-2\pi\) to \(2\pi\) for the x-axis since you are requested to graph the functions within this interval. Adjust the y-axis as well to view the entire range of function values, typically from -1 to 1 for the sine function.
04

Graph the Functions

Plot the functions together on the same coordinate system. Observe each graph carefully to compare how they differ. They should all follow the basic sine wave's shape but differ in amplitude.
05

Analyze the Effect of A

Observe how different values of \(A\) affect the graph's amplitude. - For \( A = 1 \), the graph's amplitude is 1.- For \( A = \frac{1}{2} \), the amplitude is halved, resulting in the graph not reaching above or below \( \frac{1}{2} \).- For \( A = \frac{1}{3} \), the amplitude further decreases, limiting its maximum and minimum to \( \frac{1}{3} \). The larger the value of \( A \), the taller the peaks and deeper the troughs, indicating that the amplitude of the sine function is directly proportional to the value of \( A \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
When discussing trigonometric functions, amplitude refers to the height of the wave from its middle to its peak or trough. In simpler terms, it's how "tall" or "short" the wave appears.
For the sine function, the amplitude is represented by the coefficient in front of the sine, often symbolized as "A". In the function \(y = A \sin x\), the amplitude is the absolute value of \(A\).
  • If \(A = 1\), the wave peaks at 1 and troughs at -1.
  • If \(A = \frac{1}{2}\), the wave only reaches as high as \(\frac{1}{2}\) and as low as \(-\frac{1}{2}\).
  • If \(A = \frac{1}{3}\), the highest and lowest points are \(\frac{1}{3}\) and \(-\frac{1}{3}\), respectively.
The amplitude shows how "compressed" or "stretched" the wave appears vertically. A larger \(A\) results in a taller wave, making the sine function more pronounced.
Radian Mode
When graphing trigonometric functions, especially on a graphing calculator, it is crucial to use the correct mode. This mode changes how the calculator interprets angle measures.
  • Radian mode uses radians, a unit determined by the circle's radius, to measure angles.
  • Degrees, another unit, is more common in everyday use but not particularly useful for trigonometric calculations in mathematics.
Switching to radian mode ensures the correct interpretation of the sine function over the interval \(-2\pi \leq x \leq 2\pi\). This means covering negative two full rotations of a unit circle to positive two full rotations. If you haven't switched to radian mode, the graph might look incorrect or distorted.
Sine Function
The sine function is one of the fundamental trigonometric functions. It originates from the study of right triangles but extends its utility to modeling periodic phenomena. The sine function, \(y = \sin x\), produces a wave-like pattern known as a sine wave. Its default wave repeats every \(2\pi\) radians, with a range of -1 to 1. The function is periodic with a cycle or "period" of \(2\pi\).Some key characteristics of the sine wave include:
  • Its smooth, continuous oscillation between -1 and 1.
  • A zero-crossing point at every integer multiple of \(\pi\).
  • Equidistant peaks and troughs creating a consistent pattern.;
Understanding the sine function's behavior is vital in helping predict and graph complex periodic waveforms.
Graphing Calculator
Using a graphing calculator can simplify plotting complex functions like trigonometric ones. It helps visualize the function's behavior and different transformations by displaying distinct curves on the same grid.
To graph a sine function, it's essential to:
  • Ensure the calculator is in radian mode.
  • Input each function individually; this allows you to plot them simultaneously.
  • Adjust the window settings appropriately, particularly ensuring the x-values range from \(-2\pi\) to \(2\pi\) and the y-values cover the function's amplitude adequately.
Once plotted, observe how changes in parameters like amplitude affect the graph's appearance. The graphing calculator serves as a powerful tool to intuitively understand how different mathematical transformations affect the sine function.

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Most popular questions from this chapter

Any object or quantity that is moving with a periodic sinusoidal oscillation is said to exhibit simple harmonic motion. This motion can be modeled by the trigonometric function $$ y=A \sin (\omega t) \quad \text { or } \quad y=A \cos (\omega t) $$ where \(A\) and \(\omega\) are constants. The constant \(\omega\) is called the angular frequency. A mass attached to a spring oscillates upward and downward. The displacement of the mass from its equilibrium position after \(t\) seconds is given by the function \(d=-3.5 \cos (2 \pi t)\), where \(d\) is measured in centimeters (Figure 13). a. Sketch the graph of this function for \(0 \leq t \leq 5\). b. What is the furthest distance of the mass from its equilibrium position? c. How long does it take for the mass to complete one oscillation?

Identify the amplitude and period for each of the following. Do not sketch the graph. $$ y=-10 \cos \frac{x}{10} $$

Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the amplitude for each graph. $$ y=\frac{1}{3} \sin x $$

Problems 69 through 76 will help prepare you for the next section. Use your graphing calculator to graph each family of functions for \(-2 \pi \leq x \leq 2 \pi\) together on a single coordinate system. (Make sure your calculator is set to radian mode.) What effect does the value of \(k\) have on the graph? $$ y=k+\cos x \quad \text { for } k=0, \frac{1}{2},-\frac{1}{2} $$

Make a table using multiples of \(\pi / 4\) for \(x\) to sketch the graph of \(y=\sin 2 x\) from \(x=0\) to \(x=2 \pi\). After you have obtained the graph, state the number of complete cycles your graph goes through between 0 and \(2 \pi\).
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