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Chapter 4: Problem 86

Write an equation for the function that is described by the given characteristics. A cosine curve with a period of \(4 \pi,\) an amplitude of 3 a right phase shift of \(\pi / 2,\) and a vertical translation up 2 units

Short Answer

Expert verified
The equation of the function given in the problem is \(f(x) = 3 \cos \left(\frac{1}{2}(x + \pi / 2)\right) + 2\).

Step by step solution

01

Read and Interpret the Problem

The exercise states that the function is a cosine curve with a period of \(4 \pi,\) an amplitude of 3, a right phase shiftof \(\pi / 2,\) and a vertical translation up 2 units. This information needs to be translated into an equation using the formula mentioned in the analysis. Keep in mind, the right phase shift (positive direction) will correspond to a negative phase shift in the equation.
02

Find the Amplitude, A

The amplitude of the function is given as 3. Therefore, A = 3.
03

Determine Period and Find B

The period of the function is given as \(4 \pi.\) The relation between the period \(P\) and coefficient \(B\) in the general formula is \(P = \frac{2 \pi}{|B|}\). Solving for \(B\) leads to \( B = \frac{2 \pi}{P}\) Substituting \(P = 4\pi\) gives \(B = \frac{2 \pi}{4\pi}= \frac{1}{2}\). Therefore, \(B = \frac{1}{2}\).
04

Compute Phase Shift, C

A right shift of \(\pi / 2\) corresponds to \(C = -\pi / 2\) in our equation.
05

Determine the Vertical Translation, D

A vertical translation up 2 units corresponds to \(D = 2\) in our equation.
06

Substitution and Write the Final Equation

Substituting the value of \(A, B, C,\) and \(D\) in the general equation \(f(x) = A \cos(B(x - C)) + D\), we get the equation: \(f(x) = 3\cos\left(\frac{1}{2}(x + \pi / 2)\right) + 2\)

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Most popular questions from this chapter

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