Answered: Given that f(x) = −3 cos(πx − 2) + 1,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Given that f(x) = −3 cos(πx − 2) + 1, find the following quantities: (a) The Amplitude. (b) The Period. (c) The Phase Shift. (d) One Cycle of the Graph.

Expert Solution

Step 1

Consider the given function $f (x) = - 3 \cos (πx - 2) + 1$ .

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

Where,

$|A| = amplitude, period = \frac{2 π}{B}, B ‐ cycles from 0 to 2 π C = horizontal shift D = vertical shift$

Step 2

Compare the given equation $f (x) = - 3 \cos (πx - 2) + 1$ with $y = A \cos (B x \pm C) \pm D$ .

Here, $A = - 3, B = π, C = 2 and D = 1$ .

(a) Obtain the amplitude as follows.

$\begin{array}{rcl} Amplitude & = & |A| \\ = & |- 3| \\ = & 3 \end{array}$

(b) Obtain the period as follows.

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

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