Let's go through the transformations of the function [tex]\( f(x) = \tan(Bx) \)[/tex] step by step.
1. Effect of Increasing [tex]\( B \)[/tex] on the Period:
The period of the tangent function [tex]\( \tan(x) \)[/tex] is [tex]\( \pi \)[/tex]. For the function [tex]\( f(x) = \tan(Bx) \)[/tex], the period is adjusted by the coefficient [tex]\( B \)[/tex]. Specifically, the period of [tex]\( \tan(Bx) \)[/tex] is given by [tex]\( \frac{\pi}{|B|} \)[/tex].
- When [tex]\( B \)[/tex] increases, the denominator [tex]\( |B| \)[/tex] increases, which makes the fraction smaller.
- Therefore, as [tex]\( B \)[/tex] increases, the period of the function decreases.
2. Effect of Increasing [tex]\( B \)[/tex] on the Frequency:
Frequency is the reciprocal of the period. If the period is [tex]\( T \)[/tex], then the frequency [tex]\( f \)[/tex] is [tex]\( \frac{1}{T} \)[/tex].
- Since we have already established that as [tex]\( B \)[/tex] increases, the period [tex]\( \frac{\pi}{|B|} \)[/tex] decreases.
- Hence, the frequency, being the reciprocal of the period, increases as [tex]\( B \)[/tex] increases.
3. Effect of Negative [tex]\( B \)[/tex] on the Graph:
When [tex]\( B \)[/tex] is negative, the argument of the tangent function, [tex]\( Bx \)[/tex], becomes negative, [tex]\( Bx < 0 \)[/tex].
- This reflects the graph of the function across the vertical axis because [tex]\( \tan(-x) = -\tan(x) \)[/tex].
Therefore, the correct transformations are:
- As the value of [tex]\( B \)[/tex] increases, the period of the function decreases.
- As the value of [tex]\( B \)[/tex] increases, the frequency of the function increases.
- When the value of [tex]\( B \)[/tex] is negative, the graph of the function is reflected across the vertical axis.
So, the complete statement will be:
As the value of [tex]\( B \)[/tex] increases, the period of the function decreases, and the frequency of the function increases. When the value of [tex]\( B \)[/tex] is negative, the graph of the function is reflected across the vertical axis.
