Question

Consider the function f(x) = tan 1/3 (x + pi/1) - 1
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Answer 1

Answer:

8. ☐ A Up

☑ B Down 1

☑ C Left $\displaystyle \frac{\pi}{2}$

☐ D Right

7. $\displaystyle 3\pi$

6. ☑ A x = −2π

☐ B x = −π

☐ C x = $\displaystyle \frac{\pi}{2}$

☑ D x = π

5. $\displaystyle y = -1$

Explanation:

$\displaystyle y = Atan(Bx - C) + D \\ \\ \boxed{y = tan\:(\frac{1}{3}x + \frac{\pi}{6}) - 1} \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{\pi}{B} \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow -1 \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{-\frac{\pi}{2}} \hookrightarrow \frac{-\frac{\pi}{6}}{\frac{1}{3}} \\ Wavelength\:[Period] \hookrightarrow \frac{\pi}{B} \hookrightarrow \boxed{3\pi} \hookrightarrow \frac{\pi}{\frac{1}{3}} \\ Amplitude \hookrightarrow N/A$

Here is all the information you will need. Now, what you need to know is that ALL tangent, secant, cosecant, and cotangent functions have NO amplitudes. Second, to find the period in this case, you need to take a look at the distanse between each vertical asymptote. Now, accourding to this graph, on a universal scale of $\displaystyle \frac{\pi}{4}$ with a range of $\displaystyle -2\pi\:to\:2\pi,$ the vertical asymptotes are $\displaystyle -2\pi = x\:and\:\pi = x.$ From here, you will then find the distanse between these asymptotes by simply perfourming the operation of Deduction, and doing that will give you $\displaystyle \boxed{3\pi} = \pi + 2\pi.$ So, the period of this function is $\displaystyle 3\pi.$ Now, you will instantly get the jist of the horisontal shift by looking at the above formula. Just keep in mind that the −C gives you the OPPOCITE TERMS OF WHAT THEY REALLY ARE, so you must pay cloce attention to what is given to you inside those parentheses. Finally, the midline is the centre of your graph, also known as the vertical shift, which in this case is at $\displaystyle y = -1.$

Well, that just about wraps it up. Now that everything has been explained, you should understand it better. If you still have questions, do not hesitate to comment any you have.

I am delighted to assist you at any time.