Sure! Let's analyze the function [tex]\(f(x) = \tan(Bx)\)[/tex] step by step.
1. Effect of Increasing [tex]\(B\)[/tex]:
- The general period of the tangent function [tex]\(\tan(x)\)[/tex] is [tex]\(\pi\)[/tex].
- For the function [tex]\(f(x) = \tan(Bx)\)[/tex], the period changes to [tex]\(\frac{\pi}{|B|}\)[/tex].
- As [tex]\(B\)[/tex] increases, [tex]\(|B|\)[/tex] also increases.
- Consequently, the period [tex]\(\frac{\pi}{|B|}\)[/tex] decreases as [tex]\(|B|\)[/tex] gets larger.
- The frequency of the function is the reciprocal of the period. Thus, as the period decreases, the frequency increases.
2. Effect of Negative [tex]\(B\)[/tex]:
- If [tex]\(B\)[/tex] is negative, the argument [tex]\(Bx\)[/tex] inside the tangent function changes sign, which implies a horizontal reflection of the graph of the function.
- Specifically, the graph of [tex]\(\tan(Bx)\)[/tex] for negative [tex]\(B\)[/tex] is a reflection over the y-axis compared to the graph of [tex]\(\tan(Bx)\)[/tex] for positive [tex]\(B\)[/tex].
Therefore, the completed statement describing the transformations to function [tex]\(f\)[/tex] as the value of [tex]\(B\)[/tex] changes 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.
