Question

Write the equation of a sine function that has the given characteristics. Amplitude: 2 Period: \(4 \pi\)

Short answer

The equation is \(y = 2 \,\text{sin} \left( \frac{1}{2}x \right)\).

Step-by-step solution

Identify the general form of a sine function

The general form of a sine function is given by \[y = A \,\text{sin} (B(x - C)) + D\] where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the horizontal shift, and \(D\) is the vertical shift.

Calculate the value of \(A\)

From the given characteristics, the amplitude \(A\) is 2. So, \(A = 2\).

Calculate the value of \(B\)

The period of a sine function is given by \(\frac{2\pi}{|B|}\). The problem states that the period is \(4\pi\), so we set up the equation: \[\frac{2\pi}{|B|} = 4\pi\] Solving for \(B\), we get \(|B| = \frac{1}{2}\). So, \(B = \frac{1}{2}\).

Combine the values into the general form

Substituting \(A\) and \(B\) into the general form, we get: \[y = 2 \,\text{sin} \left( \frac{1}{2}x \right)\] There is no horizontal shift \(C\) or vertical shift \(D\), so they are zero.

Key concepts

amplitude

The amplitude of a sine function describes the maximum distance the curve reaches from its central axis. It is represented by the variable \(A\) in the equation \[y = A \,\text{sin}(B(x - C)) + D\].
For instance, in the problem provided, the amplitude is specified as 2. This means that the sine wave's peaks and troughs will extend 2 units above and below the centerline, respectively.
The amplitude plays a crucial role in determining the height of the wave and gives a sense of the energy or intensity of the oscillation. When you visualize the sine curve, think of the amplitude as the “height” of the wave.

period

The period of a sine function is the length of one complete cycle of the curve. It tells you how long it takes for the function to repeat itself. In the standard sine function \[y = \text{sin}(x)\], the period is \(2\pi\), meaning after an interval of \(2\pi\) units, the curve starts to repeat.
The formula for the period in the general form \[y = A \,\text{sin}(B(x - C)) + D\] is \frac{2\pi}{|B|}\.
In the given problem, the period is \4\pi\. Solving \frac{2\pi}{|B|} = 4\pi\, we find \B = \frac{1}{2}\.
This means that the sine function completes one cycle over an interval of \4\pi\ units on the x-axis, making the curve less frequent but more stretched out compared to the standard sine function.

general form of sine function

The general form of a sine function equation is \[y = A \,\text{sin}(B(x - C)) + D\].
This equation allows various transformations of the basic sine function.

• \A\ determines the amplitude, the vertical stretch or shrink of the wave.
• \B\ affects the period, determining how frequently the waves repeat.
• \C\ introduces a horizontal shift, moving the function left or right.
• \D\ is a vertical shift, raising or lowering the entire function.

In the provided problem, we used the given amplitude \A = 2\ and calculated \B = \frac{1}{2}\ based on the period. There were no horizontal or vertical shifts, so \C\ and \D\ were zero. Hence, the equation simplified to \[y = 2 \,\text{sin}(\frac{1}{2}x)\]. This equation produces a sine wave with twice the typical height and a stretched period of \4\pi\.

trigonometry

Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. It extends to periodic functions, which are fundamental in describing wave patterns, including sine and cosine functions.
The sine function, abbreviated as \sin\, is especially important in trigonometry for modeling oscillatory behaviors like sound waves, light, and alternating currents.
Understanding sine functions involves grasping key properties like amplitude, period, and phase shifts. These characteristics allow you to transform and manipulate the function to fit real-world scenarios.

• Amplitude displays wave height.
• Period reveals how often waves repeat.
• Shifts modify wave positions.

This problem exemplifies applying these properties to write the equation \[y = 2 \,\text{sin}(\frac{1}{2}x)\], demonstrating the practical utility of trigonometry in analyzing and predicting wave behavior.