To describe the transformations to the function [tex]\( f(x) = \tan(Bx) \)[/tex] as the value of [tex]\( B \)[/tex] changes, let's break it down step-by-step:
1. Period of the Function:
- For the function [tex]\( f(x) = \tan(Bx) \)[/tex], the period is given by [tex]\( \frac{\pi}{|B|} \)[/tex].
- As the value of [tex]\( B \)[/tex] increases in absolute value, the denominator of the period formula becomes larger, making the period smaller.
- Thus, as the value of [tex]\( B \)[/tex] increases, the period of the function decreases.
2. Frequency of the Function:
- The frequency of a function is the reciprocal of the period. Therefore, for [tex]\( f(x) = \tan(Bx) \)[/tex], the frequency is [tex]\( \frac{|B|}{\pi} \)[/tex].
- As the value of [tex]\( B \)[/tex] increases in absolute value, the numerator of the frequency formula becomes larger, making the frequency larger.
- Thus, as the value of [tex]\( B \)[/tex] increases, the frequency of the function increases.
3. Graph Reflection:
- The function [tex]\( f(x) = \tan(Bx) \)[/tex] will reflect over the y-axis when [tex]\( B \)[/tex] is negative.
- Thus, when the value of [tex]\( B \)[/tex] is negative, the graph of the function reflects over the y-axis.
Putting it all together:
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 reflects over the y-axis.
Hence the dropdowns should be filled as follows:
- Period of the function: decreases
- Frequency of the function: increases
- Graph of the function: reflects over the y-axis
