Question

3.8 the tangent function
3.8 test prep

1. the graph of the function g is given in the xy-plane. if $g(x)=a\tan(bx)-20$, where a and b are constants, which of the following could be true?

i. if $a > 0$ then $b > 1$
ii. if $a > 0$ then $b < 0$
iii. if $a < 0$ then $b > 1$
iv. if $a < 0$ then $b < 0$
(a) iii only
(b) iv only
(c) i and iv only
(d) ii and iii only
graph of g

Explanation:

Step1: Analyze period of tangent function

The standard tangent function $\tan(x)$ has a period of $\pi$. From the graph, the period of $g(x)$ is $\frac{\pi}{2}$. The period of $g(x)=a\tan(bx)-20$ is $\frac{\pi}{|b|}$. Set up the equation:
$$\frac{\pi}{|b|}=\frac{\pi}{2}$$
Solve for $|b|$: $|b|=2$, so $|b|>1$.

Step2: Analyze sign of $a$ and $b$

The standard $\tan(x)$ increases from left to right. The graph of $g(x)$ decreases from left to right, so there is a reflection over the x-axis, meaning $a$ and $b$ have opposite signs:

• If $a>0$, then $b<0$
• If $a<0$, then $b>1$ (since $|b|>1$)

Step3: Match with given statements

Check each statement:

• I. If $a>0$ then $b>1$: False (needs $b<0$)
• II. If $a>0$ then $b<0$: True
• III. If $a<0$ then $b>1$: True
• IV. If $a<0$ then $b<0$: False (needs $b>1$)

Answer:

(D) II and III only