Problem 72 - Thomas Calculus

Question

The amplitude \(A\) Set the constants \(B=6, C=D=0\). \begin{equation} \begin{array}{l}{\text { a. Describe what happens to the graph of the general sine }} \\ {\text { function as } A \text { increases through positive values. Confirm your }} \\ {\text { answer by plotting } f(x) \text { for the values } A=1,5, \text { and } 9 .} \\ {\text { b. What happens to the graph for negative values of } A ?}\end{array} \end{equation}

Step-by-Step Solution

Answer

As \( A \) increases positively, the amplitude of the sine wave increases. For negative \( A \), the wave is reflected across the x-axis but retains its amplitude.

Step 1: Understanding the General Sine Function

The general form of a sine function is \( f(x) = A \sin(Bx + C) + D \). In this problem, the constants \( B = 6 \), \( C = 0 \), and \( D = 0 \) are given, which simplifies the function to \( f(x) = A \sin(6x) \). The amplitude \( A \) affects the 'height' of the wave.

Step 2: Effect of Positive Amplitude on the Graph

As \( A \) increases, the amplitude of the sine wave increases, meaning the peaks and troughs of the graph will become taller. Specifically, the maximum and minimum values will be \( A \) and \( -A \), respectively. For example, for \( A = 1 \), the amplitude is 1, so the wave oscillates between -1 and 1. At \( A = 5 \), the wave oscillates between -5 and 5, and at \( A = 9 \), it oscillates between -9 and 9.

Step 3: Effect of Negative Amplitude on the Graph

When \( A \) is negative, the effect is a reflection across the x-axis. This means the graph's peaks become troughs and vice versa, but the absolute value of \( A \) still determines the amplitude. So, for \( A = -5 \), the graph will oscillate between 5 and -5, reflected compared to \( A = 5 \).

Step 4: Graph Plotting

By plotting \( f(x) = A \sin(6x) \) for \( A = 1 \), \( A = 5 \), and \( A = 9 \), you will observe the increasing amplitude with each step. If you also plot \( A = -5 \), you will see the reflected sine wave with the same amplitude as \( A = 5 \). The larger \( |A| \) becomes, the 'taller' the oscillations become, either above or below the x-axis.

Key Concepts

Sine Wave Behavior

When we talk about sine wave behavior, we're essentially diving into the basic characteristics of a sine wave, which is a fundamental component in trigonometry. A sine wave is periodic, meaning it repeats itself at regular intervals. This periodic nature is described by various features including amplitude, frequency, and phase shift. In this case, since constants like \( B \), \( C \), and \( D \) have specific values, our focus is mainly on amplitude, denoted by \( A \).

The amplitude of a sine wave defines its height from the equilibrium position, or put simply, from the center line of the wave to the peak or the trough. As amplitude changes, it affects how 'tall' or 'short' the waves appear.

• A larger amplitude results in waves that reach higher and lower values.
• A smaller amplitude results in waves that are compressed closer to the center line.

Understanding sine wave behavior is crucial for grasping more complex concepts like graph transformations. By observing changes in these basic dynamics, students can begin to predict how other alterations to a wave's equation will manifest graphically.

Graph Transformations

Graph transformations help us understand how changes in a mathematical function affect its graphical representation. In the case of sine functions, several transformations may happen, such as vertical shifts, stretching, compressing, or reflecting.

For our specific instance where \( f(x) = A \sin(6x) \), we are primarily focused on vertical stretching and compressing. As \( A \) increases, so does the stretching of the graph vertically. This means higher peaks and lower troughs.

• For \( A = 1 \), the graph reaches 1 and -1, typical for a basic sine wave with no vertical transformation.
• Increasing \( A \) to 5 or 9 stretches the graph such that it reaches as high as 5 or 9 and as low as -5 or -9.

The graph transformation with respect to amplitude is an intuitive way to visualize how much a wave can expand or contract vertically. This fundamental understanding prepares us for more advanced graph manipulations, such as phase shifts or horizontal stretches.

Reflection Across the X-Axis

Reflection across the x-axis is an essential transformation to understand, especially in the context of negative amplitudes. When \( A \) is negative in the sine function \( f(x) = A \sin(Bx + C) + D \), the graph undergoes reflection.

This reflection transforms the graph in a manner that what are originally peaks (high points on the graph) are transformed into troughs (low points), and vice versa.

• For a positive \( A \), the graph follows the standard pattern above the x-axis.
• For a negative \( A \), say \( A = -5 \), while the absolute amplitude remains, the peaks are inverted. Hence the graph appears as if flipped upside down compared to the situation where \( A = 5 \).

Understanding this concept is pivotal for visualizing and predicting the behavior of trigonometric graphs under various conditions. It highlights how reflection inherently affects both the direction and appearance of the sine wave, creating a mirrored image across the x-axis.