Select the correct answer from each drop-down menu. Consider function [tex]$f$[/tex], where [tex]$B$[/tex] is a real number. [tex]$f(x)=\tan (B x)$[/tex] Complete the statement describing the transformations to function [tex]$f$[/tex] as the value of [tex]$B$[/tex] is changed. As the value of [tex]$B$[/tex] increases, the period of the function [ ] and the frequency of the function [ ]. When the value of [tex]$B$[/tex] is negative, the graph of the function [ ] [ ].
Answer (2)
As the value of B increases, the period of the function decreases.
As the value of B increases, the frequency of the function increases.
When the value of B is negative, the graph of the function is reflected over the y-axis.
The final answer is: decreases, increases, reflected over the y-axis.
Explanation
Problem Analysis We are given the function f ( x ) = \t \an ( B x ) , where B is a real number. We need to describe the transformations to the function as the value of B changes, focusing on the period, frequency, and symmetry.
Period Transformation The period of the standard tangent function tan ( x ) is π . When we have tan ( B x ) , the period becomes ∣ B ∣ π . As the value of B increases, the period ∣ B ∣ π decreases because we are dividing π by a larger number.
Frequency Transformation The frequency is the reciprocal of the period. Since the period of f ( x ) = tan ( B x ) is ∣ B ∣ π , the frequency is π ∣ B ∣ . As the value of B increases, the frequency π ∣ B ∣ also increases because we are multiplying π 1 by a larger number.
Symmetry Analysis The tangent function is an odd function, which means tan ( − x ) = − tan ( x ) . Therefore, when B is negative, we have f ( x ) = tan ( B x ) = tan ( − ∣ B ∣ x ) = − tan ( ∣ B ∣ x ) . This means that the graph of the function is reflected over the x-axis. However, since the tangent function is odd, the graph is unchanged.
Final Answer Therefore, as the value of B increases, the period of the function decreases, and the frequency of the function increases. When the value of B is negative, the graph of the function is reflected over the y-axis.
Examples
Understanding how changing parameters in trigonometric functions affects their period and frequency is crucial in many fields. For example, in electrical engineering, the frequency of an alternating current (AC) signal is directly related to the number of cycles it completes per second. By adjusting the parameters of a sinusoidal function, engineers can control the frequency and period of AC signals, which is essential for designing and operating electrical circuits and devices. Similarly, in music, the frequency of a sound wave determines its pitch, and musicians can manipulate frequencies to create different musical notes and harmonies.
As the value of B increases, the period of the function decreases and the frequency increases. When B is negative, the graph of the function is reflected over the y-axis.
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