Consider the given function y=-8cos6x+2π3+4. Note that, the general form of the cosine function is…

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Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterA: Appendix

SectionA.3: Inequalities

Problem 11E

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Step 1

Consider the given function $y = - 8 \cos (6 x + \frac{2 π}{3}) + 4$ .

Note that, the general form of the cosine function is $y = A \cos (B x \pm C) \pm D$ .

Where, $|A|$ is amplitude, B - cycles from 0 to $2 π$ . That implies, $period = \frac{2 π}{B}$ .

C - horizontal shift (known as phase shift when B = 1), and D - vertical shift (displacement).

Step 2

Compare the given function $y = - 8 \cos (6 x + \frac{2 π}{3}) + 4$ with $y = A \cos (B x \pm C) \pm D$ .

$A = - 8, B = 6, C = \frac{2 π}{3}, and D = 4$ .

That implies,

$\begin{array}{rcl} Amplitude & = & |A| \\ = & |- 8| \\ = & 8 \end{array}$

$\begin{array}{rcl} Period & = & \frac{2 π}{6} \\ = & \frac{π}{3} \end{array}$

$\begin{array}{rcl} horizontal shift & = & C \\ = & \frac{2 π}{3} \end{array}$

$\begin{array}{rcl} vertical shift & = & D \\ = & 4 \end{array}$

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