practice 4-6-15 write an equation that represents all asymptotes of the graph of f in the xy - plane. 1…

Explanation:

Step1: Recall tangent - asymptote formula

The tangent function (y = \tan(x)) has vertical asymptotes at (x=(n+\frac{1}{2})\pi), where (n\in\mathbb{Z}).

Step2: Find asymptotes for (f(\theta)=\tan(9\theta))

Set (9\theta=(n + \frac{1}{2})\pi), then solve for (\theta). We get (\theta=\frac{(2n + 1)\pi}{18}), (n\in\mathbb{Z}).

Step3: Find asymptotes for (f(\theta)=\tan(\frac{\theta}{4}))

Set (\frac{\theta}{4}=(n+\frac{1}{2})\pi), then solve for (\theta). Multiply both sides by 4, we have (\theta=(4n + 2)\pi), (n\in\mathbb{Z}).

Answer:

1. (\theta=\frac{(2n + 1)\pi}{18},n\in\mathbb{Z})
2. (\theta=(4n + 2)\pi,n\in\mathbb{Z})