Let's analyze the function [tex]\( f(x) = \tan(Bx) \)[/tex] and describe how the transformations occur as the value of [tex]\( B \)[/tex] changes.
1. Period of the Function:
The period of the tangent function [tex]\( \tan(Bx) \)[/tex] is given by [tex]\( \frac{\pi}{|B|} \)[/tex]. As [tex]\( B \)[/tex] increases, the denominator becomes larger, resulting in a smaller period. Therefore, as the value of [tex]\( B \)[/tex] increases, the period of the function decreases.
2. Frequency of the Function:
The frequency of the function is the reciprocal of the period. Hence, the frequency is given by [tex]\( \frac{|B|}{\pi} \)[/tex]. As [tex]\( B \)[/tex] increases, the frequency becomes larger. Thus, as the value of [tex]\( B \)[/tex] increases, the frequency of the function increases.
3. When [tex]\( B \)[/tex] is Negative:
When [tex]\( B \)[/tex] is negative, the function [tex]\( \tan(Bx) \)[/tex] experiences a reflection over the y-axis because replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the function [tex]\( \tan(x) \)[/tex] results in [tex]\( \tan(-x) = -\tan(x) \)[/tex]. Therefore, when the value of [tex]\( B \)[/tex] is negative, the graph of the function is a reflection over the y-axis.
Putting it all together, the completed statement is:
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 a reflection over the y-axis.
