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Chapter 1: Problem 4

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(j(u)=7 \sin (2 u+\pi)-6\)

Short Answer

Expert verified
Amplitude: 7, Period: \(\pi\), Average value: -6, Horizontal shift: \(-\frac{\pi}{2}\) units right.

Step by step solution

01

Identify the Amplitude

The amplitude of a sine function, given by the form \(a \sin(bx + c) + d\), is \(|a|\). In the function \(j(u) = 7 \sin(2u + \pi) - 6\), the amplitude \(|a|\) is \(|7| = 7\).
02

Determine the Period

The period of a sine function is calculated using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of the angle variable inside the sine function. Here, \(b = 2\), so the period is \(\frac{2\pi}{2} = \pi\).
03

Find the Average Value

The average value of the sine function \(a \sin(bx + c) + d\) is the value of \(d\). In this function, the average value is \(-6\).
04

Calculate the Horizontal Shift

The horizontal shift is determined from the phase shift \(\frac{-c}{b}\), where \(c\) is the constant added to \(bx\) in the sine function. Here, \(c = \pi\), so the horizontal shift is \(\frac{-\pi}{2}\). This means the function is shifted \( \frac{-\pi}{2} \) units to the right.

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Key Concepts

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

Amplitude
Amplitude is a crucial concept when dealing with trigonometric functions like sine and cosine. It measures the distance from the middle of the wave to its peak. In mathematical terms, for sinusoidal functions such as \( a \sin(bx + c) + d \), amplitude is denoted by \(|a|\).
  • Amplitude represents the height of the wave above or below its midline.
  • In the given function \( j(u) = 7 \sin(2u + \pi) - 6 \), the amplitude is \( |7| = 7 \).
  • The amplitude informs us how "tall" or "short" the wave is, contributing to the wave's energy.
Understanding amplitude helps in predicting the maximum displacement from the equilibrium. It is fundamentally significant in various real-world applications, including sound waves, electricity, and even tides.
Period
The period of a trigonometric function determines how long it takes to repeat a full cycle. It essentially tells us the amount of input needed for the function to start over. For sine and cosine functions, calculating the period involves the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\).
  • In the function \( j(u) = 7 \sin(2u + \pi) - 6 \), the coefficient \(b\) is \(2\).
  • This leads to a period calculation of \(\frac{2\pi}{2} = \pi\).
  • A period of \(\pi\) implies the function rewinds after every \(\pi\) units along the horizontal axis.
The concept of period is vital in various fields, from analyzing repetitive behaviors in nature to designing circuits that oscillate at specific frequencies.
Phase Shift
Phase shift indicates the horizontal displacement of a trigonometric function. It defines how far the function has shifted to the right or left compared to the usual position. For the function \( a \sin(bx + c) + d \), the phase shift is calculated as \( \frac{-c}{b} \).
  • In the function \( j(u) = 7 \sin(2u + \pi) - 6 \), "\(c\)" is \(\pi\).
  • The phase shift, therefore, is \( \frac{-\pi}{2} \).
  • This indicates that the sine wave is shifted \( \frac{-\pi}{2} \) units to the right.
Phase shift plays a substantial role in modeling scenarios where timing is crucial, such as in engineering and physics. It helps synchronize waves or signals to create desired outcomes.

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Most popular questions from this chapter

Input and output notation is given for two functions. Determine whether the pair of functions can be combined by function composition. If so, then a. draw an input/output diagram for the new function. b. write a statement for the new function complete with function notation and input and output units and descriptions. The average number of customers in a restaurant on a Saturday night \(t\) hours since 4 P.M. is \(C(t)\) customers. The average amount of tips generated by \(c\) customers is \(P(c)\) dollars.

Bankruptcy Filings Total Chapter 12 bankruptcy filings between 1996 and 2000 can be modeled as $$ B(t)=-83.9 t+1063 \text { filings } $$ where \(t\) is the number of years since \(1996 .\) (Source: Based on data from Administrative Office of U.S. Courts) a. Give the rate of change of \(B\). Include units. b. Draw a graph of the model. Label the graph with its slope. c. Evaluate \(B(0)\). Give a sentence of practical interpretation for the result. d. Calculate the number of bankruptcy filings in \(1999 .\) Is this interpolation or extrapolation? Explain.

pH Levels The pH of a solution, measured on a scale from 0 to \(14,\) is a measure of the acidity or alkalinity of that solution. Acidity/alkalinity \((\mathrm{pH})\) is a function of hydronium ion \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration. The table shows the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration and associated \(\mathrm{pH}\) for several solutions. $$ \begin{array}{|c|c|c|} \hline \text { Solution } & \begin{array}{c} \mathrm{H}_{3} 0^{+} \\ \text {(moles per liter) } \end{array} & \mathrm{pH} \\ \hline \text { Cow's milk } & 3.98 \cdot 10^{-7} & 6.4 \\ \hline \text { Distilled water } & 1.0 \cdot 10^{-7} & 7.0 \\ \hline \text { Human blood } & 3.98 \cdot 10^{-8} & 7.4 \\ \hline \text { Lake Ontario water } & 1.26 \cdot 10^{-8} & 7.9 \\ \hline \text { Seawater } & 5.01 \cdot 10^{-9} & 8.3 \\ \hline \end{array} $$ a. Find a log model for \(\mathrm{pH}\) as a function of the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration. b. What is the \(\mathrm{pH}\) of orange juice with \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration \(1.56 \cdot 10^{-3} ?\) c. Black coffee has a \(\mathrm{pH}\) of \(5.0 .\) What is its concentration of \(\mathrm{H}_{3} \mathrm{O}^{+} ?\) d. A pH of 7 is neutral, a pH less than 7 indicates an acidic solution, and a pH greater than 7 shows an alkaline solution. What \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration is neutral? What \(\mathrm{H}_{3} \mathrm{O}^{+}\) levels are acidic and what \(\mathrm{H}_{3} \mathrm{O}^{+}\) levels are alkaline?

Walking Speed The relation between the average walking speed and city size for 36 selected cities of population less than 1,000,000 has been modeled as \(v(s)=0.083 \ln p+0.42\) meters per second where \(p\) is the population of the city. (Source: Based on data in Bornstein. International Journal of Psychology.1979.) a. What is the average walking speed in a city with a population of \(1,000,000 ?\) b. Does a city with a population of 1000 have a faster or slower average walking speed than a city with a population of \(100,000 ?\) c. Give possible reasons for the results of the walking speed research.

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad p(x)=\sin (2.2 x+0.4)+0.7\)
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