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Chapter 1: Problem 67

For $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$, identify \(A, B, C,\) and \(D\) for the sine functions and sketch their graphs. $$y=-\frac{2}{\pi} \sin \left(\frac{\pi}{2} t\right)+\frac{1}{\pi}$$

Short Answer

Expert verified
A = -\frac{2}{\pi}, B = 4, C = 0, D = \frac{1}{\pi}.

Step by step solution

01

Identify Coefficient A

In the given function, the coefficient in front of the sine function determines the amplitude. For the function \( y = -\frac{2}{\pi} \sin \left(\frac{\pi}{2} t\right) + \frac{1}{\pi} \), the amplitude \( A \) is \( -\frac{2}{\pi} \). This value is taken directly from the coefficient of the sine term.
02

Identify Coefficient B

The coefficient within the sine function determines the period of the sine wave. Here, it is represented by \( \frac{2\pi}{B} = \frac{\pi}{2} \). Solving for \( B \), we have \( B = 4 \), since \( \frac{2\pi}{4} = \frac{\pi}{2} \).
03

Identify Phase Shift C

The phase shift is represented by the \( (x-C) \) term in the generic function. In the function \( y = -\frac{2}{\pi} \sin \left(\frac{\pi}{2} t\right) + \frac{1}{\pi} \), there is no \( C \) present. Therefore, the phase shift \( C \) is 0.
04

Identify the Vertical Shift D

The vertical shift \( D \) is the constant term added or subtracted from the sine function. Here, it is represented by \( \frac{1}{\pi} \), which is the constant added to the sine function. Thus, \( D = \frac{1}{\pi} \).
05

Sketch the Graph

To sketch the graph of the function, note the following characteristics: the amplitude is \( -\frac{2}{\pi} \), indicating a vertical stretch and a reflection over the x-axis; the period is \( 4 \); there is no phase shift, and there is a vertical shift of \( \frac{1}{\pi} \). The graph will oscillate between \( \frac{1}{\pi} + \frac{2}{\pi} \) and \( \frac{1}{\pi} - \frac{2}{\pi} \) with a period of 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
The amplitude of a trigonometric function, particularly sine or cosine, dictates how much the function extends above and below the central axis (or midline) of the wave.This might sound complex, but it's quite simple:
  • The amplitude is the absolute value of the coefficient in front of the sine or cosine function.
  • In mathematical terms, if you have a function of the form \( y = A \sin(x) \) or \( y = A \cos(x) \), the amplitude is \( |A| \).
In the function \( y = -\frac{2}{\pi} \sin \left(\frac{\pi}{2} t \right) + \frac{1}{\pi} \), the amplitude is \( \left| -\frac{2}{\pi} \right| = \frac{2}{\pi} \).
Do notice that the negative sign indicates a reflection across the x-axis, but does not affect the measure of amplitude.Understanding amplitude helps predict how far and in what direction the wave will oscillate from its midline position.
Period
The period of a sine or cosine function tells us how long it takes for the waveform to complete one full cycle and start repeating.This is a fundamental concept because it determines the horizontal stretch of the graph.
  • The formula for finding the period of a sine or cosine function is \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) (or \( t \) in this problem).
In our exercise, the sine function is \( y = -\frac{2}{\pi} \sin \left(\frac{\pi}{2} t \right) \). Solving \( \frac{2\pi}{B} = \frac{\pi}{2} \) for \( B \), we find that \( B = 4 \).
This means that the period is 4, implying the wave pattern repeats every 4 units along the x-axis.
Phase Shift
Phase shift refers to the horizontal shifting of a trigonometric function's graph left or right.This shift is determined by the \( (x-C) \) or \( (t-C) \) factor within the sine or cosine function.
  • If we have the form \( A \sin(B(x-C)) \), the graph shifts \( C \) units.
  • A positive \( C \) moves the graph to the right, and a negative \( C \) moves it to the left.
In the given function \( y = -\frac{2}{\pi} \sin \left(\frac{\pi}{2} t \right) + \frac{1}{\pi} \), there is no \( C \) present, so the phase shift is 0.
This means the graph remains in its standard position without any horizontal movement.
Vertical Shift
The vertical shift of a sine or cosine wave is determined by any constant added outside the trigonometric function.This shift moves the entire graph up or down along the y-axis.
  • If the equation is \( y = A \sin(Bx) + D \), then \( D \) indicates the vertical shift.
  • A positive \( D \) shifts the graph upward, while a negative \( D \) shifts it downward.
In our example, the function is \( y = -\frac{2}{\pi} \sin \left(\frac{\pi}{2} t \right) + \frac{1}{\pi} \), where \( D = \frac{1}{\pi} \).
This means the entire graph is shifted \( \frac{1}{\pi} \) units upward, so its center is no longer at the x-axis but at \( y = \frac{1}{\pi} \).

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Most popular questions from this chapter

Use a graphing utility to find the regression curves specified. The table shows the average weight for men of medium frame based on height as reported by the Metropolitan Life Insurance Company (1983). $$\begin{array}{cc|cc}\hline \text { Height (cm) } & \text { Weight (kg) } & \text { Height (cm) } & \text { Weight (kg) } \\ \hline 157.5 & 61.7 & 177.5 & 71.2 \\\160 & 62.6 & 180 & 72.6 \\\162.5 & 64 & 182.5 & 74.2 \\\165 & 64.2 & 185 & 75.8 \\\167.5 & 65.8 & 187.5 & 77.6 \\\170 & 67.1 & 190 & 79.2 \\\172.5 & 68.5 & 192.5 & 81.2 \\\175 & 69.9 & & \\\\\hline\end{array}$$ a. Make a scatterplot of the data. b. Find and plot a regression line, and superimpose the line on the scatterplot. c. Does the regression line reasonably capture the trend of the data? What weight would you predict for a male of height \(2 \mathrm{m} ?\)

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Find the function values. $$\sin ^{2} \frac{3 \pi}{8}$$

a. Graph the functions \(f(x)=x / 2\) and \(g(x)=1+(4 / x)\) together to identify the values of \(x\) for which $$\frac{x}{2}>1+\frac{4}{x}.$$ b. Confirm your findings in part (a) algebraically.

For $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$, identify \(A, B, C,\) and \(D\) for the sine functions and sketch their graphs. $$y=\frac{1}{2} \sin (\pi x-\pi)+\frac{1}{2}$$
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