Problem 51 - Algebra 2

Question

Write the equations of three sine functions with the same amplitude that have periods of \(2,3,\) and \(4 .\) Then sketch all three graphs on the same coordinate axes.

Step-by-Step Solution

Answer

The three sine functions with different periods are \(y_1 = A*sin(\pi x)\), \(y_2 = A*sin(\frac{2\pi x}{3})\), \(y_3 = A*sin(\frac{\pi x}{2})\), where A is the common amplitude. To plot these, select a range of x-values and calculate the corresponding y-values for each function, then plot all points on the same set of axes.

Step 1: Write sine functions for given periods

Given the periods 2, 3, and 4 for each respective sine function and all have same amplitude A, we can substitute these into the general form of a sine function, yielding: \(y_1 = A*sin(\frac{2\pi x}{2})\), \(y_2 = A*sin(\frac{2\pi x}{3})\), \(y_3 = A*sin(\frac{2\pi x}{4})\) which simplifies to \(y_1 = A*sin(\pi x)\), \(y_2 = A*sin(\frac{2\pi x}{3})\), \(y_3 = A*sin(\frac{\pi x}{2})\)

Step 2: Plot sine functions

To plot the functions on the same graph, a value for the amplitude A must be chosen. It can be any number, but for simplicity let's choose A = 1. The x-values ranges could also be any values, but it's easiest to use the range \(-2\pi \leq x \leq 2\pi\) to cover at least one complete period of each function. Now, using the equations from Step 1, the y-values can be calculated for each x-value and plotted on the same set of axes.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is defined as a periodic function that relates to the angles of a triangle to the length of its sides. In mathematical terms, the sine function can be written as \( y = A \sin(Bx + C) + D \), where the coefficients \( B, C, \) and \( D \) control the transformation of the function, and \( A \) represents the amplitude.

• Smooth Wave-like Patterns: The sine function produces a smooth, continuous wave-like curve when graphed.
• Oscillation: It oscillates above and below an axis, typically the x-axis, showing the repeating nature of the function.
• Repetitive Behavior: The shape of the graph repeats at regular intervals known as "periods".

Understanding the sine function is crucial as it sets the foundation for analyzing more complex trigonometric functions used in various real-world applications, such as physics and engineering.

Amplitude

Amplitude represents the height of the wave or trigonometric function from its central axis to its peak or trough. In the context of a sine function, it is the distance from the midline of the graph to the maximum (or minimum) point. The amplitude can always be found by the absolute value of \( A \), where the equation of the sine wave is \( y = A \sin(Bx + C) + D \).

• Defining the Peak: If \( A = 1 \), the wave's peak and trough extend 1 unit above and below the midline, respectively.
• Impact on Graph: A higher amplitude means a taller wave, while lower values result in a flatter wave.
• Amplitude as Constant: In problems about these sine waves, it's crucial to ensure all comparisons are made with the same amplitude for a consistent analysis.

The amplitude informs us of the energy or intensity of the wave and holds practical importance in fields like audio engineering, where it relates to sound volume.

Period

The period of a sine function refers to the distance along the x-axis required for the graph to complete one full cycle of its pattern. Mathematically, the period \( T \) can be calculated using the formula \( T = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the sine function's argument \( \sin(Bx) \).

• Determines Cycle Length: A smaller period results in more frequent repetitions along the x-axis, indicating a quicker oscillation.
• Longer Periods: Conversely, a larger period means fewer oscillations over the same x-axis length.
• Independent of Amplitude: The period is not affected by changes in amplitude, making it vital to isolate period effects in graph analysis.

Knowing a function's period allows for predictions of wave behavior over time, useful in fields such as signal processing and animations.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as the sine function, provides a visual representation of how these functions behave. The graph of a sine function is a continuous wave that repeats every period, while having peaks and troughs determined by the amplitude.

• Scaling and Shifting: Sine function graphs can be scaled horizontally by their periods, and vertically by their amplitudes.
• Coordinate Axes: For effective graphing, select an appropriate x-range; for example, \( -2\pi \leq x \leq 2\pi \) to capture full cycles.
• Plotting Points: While creating graphs, calculating multiple y-values at evenly spaced x-values helps in drawing smooth curves.

Graphing is a fundamental part of understanding trigonometric functions, enabling identification of key characteristics like amplitude, period, and phase shift through visual inspection. It's readily applied when assessing wave phenomena, such as sound waves or electrical signals.