9. the graph of the function f is given in the xy - plane. if f(x)=a tan(bx), where a and b are constants…

Explanation:

Step1: Analyze the amplitude

The general form of the tangent - function is (y = a\tan(bx)). The sign of (a) determines the reflection of the graph of (y=\tan(x)) about the (x) - axis. If (a>0), the graph has the same general shape as (y = \tan(x)); if (a < 0), the graph is reflected about the (x) - axis. The given graph of (y=a\tan(bx)) is reflected about the (x) - axis compared to the standard graph of (y = \tan(x)), so (a<0).

Step2: Analyze the period

The period of the tangent function (y=\tan(bx)) is given by (T=\frac{\pi}{|b|}). The standard period of (y = \tan(x)) is (\pi). For the function (y=\tan(bx)), if (|b|>1), the period is less than (\pi), and if (|b|<1), the period is greater than (\pi). The period of the given function (y = a\tan(bx)) is (4\pi). Since (T=\frac{\pi}{|b|}=4\pi), we can solve for (|b|): [|b|=\frac{1}{4}<1]

Answer:

D. (a < 0) and (b<1)