/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 55

Sketch the graph of the function. (Include two full periods.) $$y=2+\frac{1}{10} \cos 60 \pi x$$

Short Answer

Expert verified
The graph of the function \(y=2+\frac{1}{10} \cos 60 \pi x\) is a cosine curve with a period of \( \frac{1}{30} \), oscillating between 1.9 and 2.1 with the center at 2. There are no phase shifts.

Step by step solution

01

Determine the Parameters

The function is of the form \(y=a+b \cos c (x-d)\), where:\n- \(a\) is the vertical shift, \n- \(b\) is the amplitude, \n- \(c\) is related to period of the function: the period is given by \(\frac{2\pi}{|c|}\), \n- \(d\) is the phase shift. For our function \(y=2+\frac{1}{10} \cos 60 \pi x\) we have:\n- \(a = 2\) (thus a vertical shift of 2 units upwards), \n- \(b= \frac{1}{10}\) (thus the amplitude is \(\frac{1}{10}\)), \n- \(c=60\pi\) (thus the period is \(\frac{2\pi}{60\pi}=\frac{1}{30}\)) and \n- there is no phase shift as the \(x\) term doesn't have a subtraction or addition.
02

Plot Key Points

Use the phase shift, amplitude, period and vertical shift to draw key points. At \(x=0\), the value of the cosine function is 1, so the first point on our graph is \((0, a + b) = (0, 2+ \frac{1}{10}) = (0, 2.1)\). Then we mark the next peaks and troughs by adding/subtracting the amplitude with the period \( \frac{1}{30} \) for each consecutive x-value.
03

Sketch the Graph

Join the key points to draft the cosine curve. The graph spans vertically between (2-0.1) and (2+0.1), i.e., from 1.9 to 2.1, oscillating every \( \frac{1}{30} \) units.

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