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Chapter 1: Problem 5

Graph the function. What is the amplitude and period? $$y=5-\sin 2 t$$

Short Answer

Expert verified
Amplitude = 1; Period = \( \pi \).

Step by step solution

01

Identify the Basic Trigonometric Function

The given function is of the form \(y = a + b \sin(ct)\). Here, the trigonometric function used is the sine function \( \sin(2t) \). This informs us that the basic form is a sinusoidal wave.
02

Find the Amplitude

For a function of the form \(y = a + b \sin(ct)\), the amplitude is the absolute value of the coefficient of the sine function. Here, the function is \(y = 5 - \sin(2t) = 5 + (-1)\sin(2t)\), so the amplitude is \( |b| = |-1| = 1 \).
03

Determine the Vertical Shift

The vertical shift in the function \(y = a + b \sin(ct)\) is given by \(a\). In this function, \(a = 5\). This means the entire sine wave is shifted 5 units upwards.
04

Calculate the Period

The period of a sine function \(\sin(ct)\) is calculated as \( \frac{2\pi}{c} \). In this function, \(c = 2\), so the period is \( \frac{2\pi}{2} = \pi \). This means one complete cycle of the wave occurs from \(0\) to \(\pi\).
05

Graph the Function

Start by plotting the baseline \(y = 5\), which is the vertical shift. Then plot the sine wave \(-\sin(2t)\) with amplitude 1, having peaks and troughs at \( y = 6 \) and \( y = 4 \) respectively. Mark the key points: the sine reaches maximum at \(t = \frac{\pi}{2}\), crosses the baseline at integer multiples of \(\pi\), and reaches minimum at odd multiples of \(\frac{3\pi}{2}\). Sketch the periodic continuation of this cycle along the horizontal axis.
06

Confirm the Characteristics

Review the graph to ensure it matches the function \(y = 5 - \sin(2t)\). The amplitude of 1 suggests oscillations of 1 unit above and below the baseline. The period of \(\pi\) confirms that these oscillations repeat every \(\pi\) units.

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Key Concepts

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

Amplitude
In trigonometric functions, amplitude refers to the measure of how far the wave peaks and troughs from its central or midline position. In simple terms, it is half the distance from the maximum to the minimum values of the wave. For the function form \(y = a + b \sin(ct)\), the amplitude is given by the absolute value of coefficient \(b\).
In our situation, the function \(y = 5 - \sin(2t)\) can be seen as \(y = 5 + (-1)\sin(2t)\), hence the amplitude is \(|-1| = 1\).
  • This value of 1 indicates that the wave extends 1 unit above and below its midline.
  • The amplitude fundamentally affects the height of the peaks and the depth of the troughs.
An understanding of amplitude helps in predicting the extreme points of a trigonometric graph.
Period
The period of a trigonometric function determines the length of one complete wave cycle. This is crucial in understanding how the wave repeats itself over a given span. For a sine function \(\sin(ct)\), the period is calculated as \(\frac{2\pi}{c}\).
In the function \(y = 5 - \sin(2t)\), \(c\) is 2, giving a period of \( \frac{2\pi}{2} = \pi \).
  • This means every \(\pi\) units along the horizontal axis, the wave pattern completes a full cycle.
  • The period helps determine the frequency of occurrences of peaks and zero crossings on a graph.
With a period of \(\pi\), one can anticipate where the function will return to its starting point along its horizontal trajectory.
Sinusoidal Wave
A sinusoidal wave is a mathematical curve that describes a smooth repetitive oscillation, commonly represented by sine and cosine functions. These waves are prevalent in numerous natural phenomena like sound and light.
For the given function \(y = 5 - \sin(2t)\), we recognize a sinusoidal pattern typical of a sine wave. This sinusoidal graph moves fluidly through its maximum and minimum points in a uniform sequence.
  • The structure suggests balanced oscillation around the central axis.
  • Each cycle consists of a rise to peak, a fall to trough, and a recovery back to baseline.
The sinusoidal wave helps visualize how trigonometric functions interpret periodic changes and can be scaled and shifted through modifications of their parameters.
Vertical Shift
The vertical shift of a trigonometric function indicates how far the entire waveform is moved up or down on the graph. For functions of the form \(y = a + b \sin(ct)\), this shift corresponds to the value of \(a\).
In our function \(y = 5 - \sin(2t)\), the vertical shift is \(a = 5\). Thus, the whole graph of \(-\sin(2t)\) is shifted 5 units upward.
  • This shift alters the baseline from which the amplitude is measured.
  • It modifies the central "zero point" of the trigonometric wave.
Understanding the vertical shift is essential for properly orienting a trigonometric graph within a coordinate system.

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Most popular questions from this chapter

Table 1.30 gives data for the linear demand curve for a product, where \(p\) is the price of the product and \(q\) is the quantity sold every month at that price. Find formulas for the following functions. Interpret their slopes in terms of demand. (a) \(\quad q\) as a function of \(p\) (b) \(\quad p\) as a function of \(q\) $$\begin{array}{c|c|c|c|c|c}\hline p \text { (dollars) } & 16 & 18 & 20 & 22 & 24 \\\\\hline q \text { (tons) } & 500 & 460 & 420 & 380 & 340 \\\\\hline\end{array}$$

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$2 P=P e^{0.3 t}$$

A quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{b} a^{t}\) to: (a) Find values for the parameters \(a\) and \(P_{0}\). (b) State the initial quantity and the percent rate of growth or decay. \(P=320\) when \(t=5\) and \(P=500\) when \(t=3\)

In Problems \(29-30,\) a quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{0} e^{k t}\) to: (a) Find values for the parameters \(k\) and \(P_{0}\). (b) State the initial quantity and the continuous percent rate of growth or decay. \(P=40\) when \(t=4\) and \(P=50\) when \(t=3\)

Most breeding birds in the northeast US migrate elsewhere during the winter. The number of bird species in an Ohio forest preserve oscillates between a high of 28 in June and a low of 10 in December. \(^{97}\) (a) Graph the number of bird species in this preserve as a function of \(t,\) the number of months since June. Include at least three years on your graph. (b) What are the amplitude and period of this function? (c) Find a formula for the number of bird species, \(B\), as a function of the number of months, \(t\) since June.
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