/*! 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 39 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 39

Sketch the graph of the function. (Include two full periods.) $$y=5 \sin x$$

Short Answer

Expert verified
The graph of the function \(y = 5 \sin x\) is a wave that oscillates between -5 and 5, with a period of \(2\pi\) units. It starts at the origin (0,0), peaks at \((\pi/2, 5)\), crosses back down through the x-axis at \((\pi, 0)\), troughs at \((3\pi/2, -5)\), then returns to the x-axis at \((2\pi, 0)\). It then repeats this pattern for the second period.

Step by step solution

01

Identifying the Characteristics

The function \(y = 5 \sin x\) is a sine function with amplitude 5 and period \(2\pi\). The amplitude is the distance from the maximum (peak) or minimum (trough) of the wave to the horizontal axis, in this case, the y-value will range from -5 to 5. The period, here \(2\pi\), is the length of one complete cycle of the function. There is no phase shift or vertical shift, meaning the function starts at 0 and its midline is on the x-axis.
02

Sketching the First Period

Start by plotting the point (0, 0) because the sine function begins at the origin. Then, remember that the sine function reaches its maximum at \(\pi/2\), returns back to 0 at \(\pi\), reaches its minimum at \(3\pi/2\) and returns back to 0 at \(2\pi\). Thus, we can plot the points \((0, 0)\), \((\pi/2, 5)\), \((\pi, 0)\), \((3\pi/2, -5)\), \((2\pi, 0)\). Draw a smooth curve through these points to complete the sketch of the first period.
03

Sketching the Second Period

The second period of the function will exactly mirror the first period. Therefore, plot the points \((2\pi, 0)\), \((5\pi/2, 5)\), \((3\pi, 0)\), \((7\pi/2, -5)\), \((4\pi, 0)\) and draw a smooth curve through these points.

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Most popular questions from this chapter

Angle of Elevation The height of an outdoor basketball backboard is \(12 \frac{1}{2}\) feet, and the backboard casts a shadow \(17 \frac{1}{3}\) feet long. A. Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. B. Use a trigonometric function to write an equation involving the unknown angle of elevation. C. Find the angle of elevation of the sun.

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For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{64} \sin 792 \pi t$$

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