/*! 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 3 Graph the function. $$ f(x)=... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 3

Graph the function. $$ f(x)=-\sin x $$

Short Answer

Expert verified
The graph of \( f(x) = -\sin x \) is a vertical reflection of \( \sin x \) across the x-axis.

Step by step solution

01

Identify the Function Characteristics

The given function is \( f(x) = -\sin x \). The function is a transformation of the basic sine function \( \sin x \). It is vertically reflected across the x-axis, which means each y-value of \( \sin x \) is multiplied by -1.
02

Determine Key Points to Plot

The sine function has key points at intervals of \( \frac{\pi}{2} \). For \( \sin x \): \( \sin 0 = 0 \), \( \sin \frac{\pi}{2} = 1 \), \( \sin \pi = 0 \), \( \sin \frac{3\pi}{2} = -1 \), \( \sin 2\pi = 0 \). For \( -\sin x \), these points become: \( (0, 0) \), \( \left(\frac{\pi}{2}, -1\right) \), \( (\pi, 0) \), \( \left(\frac{3\pi}{2}, 1\right) \), \( (2\pi, 0) \).
03

Sketch the Graph

On a set of axes, plot the key points found in Step 2. Start at the origin \((0, 0)\), then plot the points \( \left(\frac{\pi}{2}, -1\right), (\pi, 0), \left(\frac{3\pi}{2}, 1\right), (2\pi, 0) \). Connect these points with a smooth, continuous curve to complete one period of the function.
04

Identify the Period and Amplitude

The period of the function \(-\sin x\) is \(2\pi\), the same as \(\sin x\). The amplitude, or the distance from the midline of the graph to a peak or trough, is 1. However, because this is a reflection, the peaks are \( -1 \) and troughs are \( 1 \), opposite from the regular \( \sin x \).
05

Repeat the Pattern

To graph more than one period, extend the pattern formed in Step 3 both to the left and right, using the same interval \(2\pi\) to repeat the key points that were plotted.

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Key Concepts

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

Sine function
The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. It is commonly encountered in mathematics and plays a vital role in various applications such as physics, engineering, and music. When you graph the basic sine function, it typically has a smooth wave-like shape known as a sinusoidal curve.
The wave oscillates above and below the x-axis forming peaks and troughs. For the basic sine wave, it starts at the origin (0,0), peaks at \(x = \frac{\pi}{2}\), returns to the x-axis at \(x = \pi\), reaches a trough at \(x = \frac{3\pi}{2}\), and completes one full cycle back at \(x = 2\pi\).
  • The sine function is periodic, meaning it repeats itself over regular intervals.
  • In one complete cycle, the sine wave covers an interval of \([0, 2\pi]\).
  • Its graph illustrates the smooth, continuous variation over this interval.
Graphing transformations
Graphing transformations involve shifting, stretching, or reflecting functions in a systematic way. In the case of \(-\sin x\), it is helpful to understand that this is more than just a sine wave:
When you graph \(-\sin x\), you are reflecting the standard sine wave across the x-axis. This means for the function \(-\sin x\), wherever \(\sin x\) would have been above the x-axis, \(-\sin x\) is below and vice versa.
  • Reflecting a graph in this way simply takes all y-values of the original function and multiplies them by -1.
  • This forms the essential transformation to consider when graphing \(-\sin x\).
  • The overall shape will still be a sinusoidal curve, just flipped vertically.
Thus, when plotting \(-\sin x\), remember: a hill becomes a valley and a valley becomes a hill!
Function characteristics
The characteristics of a function are its distinct features. For \(-\sin x\), understanding its function characteristics is crucial for accurately sketching or analyzing the graph.
A key attribute of the sine wave, including \(-\sin x\), is symmetry. This function is symmetric about the origin, meaning that if you rotate the graph 180 degrees, it looks the same.
Additional characteristics for \(-\sin x\):
  • It is an odd function, a property that describes the graph's symmetry. For any input value \(x\), \(-\sin(-x) = \sin x\).
  • The sine wave is continuous and smooth, which makes it easy to connect plotted points.
  • The intercepts on the x-axis are at intervals of \(n\pi\), where \(n\) is an integer.
Period
The period of a trigonometric function refers to the interval over which the function completes one full cycle before repeating. For the standard sine function \( \sin x \), the period is \(2\pi\). This means that its wave pattern repeats every \(2\pi\) units along the x-axis.
When considering transformations like \(-\sin x\), the period remains unaffected. The reflection only changes the vertical orientation but does not alter the horizontal length of one cycle. Therefore, the period for \(-\sin x\) is still \(2\pi\).
  • Regularly occurring intervals make graphing more than one period straightforward.
  • On a graph, you'd extend the wave horizontally in both directions, repeating the pattern every \(2\pi\).
  • Understanding the period is essential for identifying how the wave continues indefinitely.
Amplitude
The amplitude of a sine wave refers to the maximum height from the wave's midline鈥攗sually the x-axis鈥攖o the peak of the wave. For the standard sine wave, the amplitude is 1.
For \(-\sin x\), the concept of amplitude helps you understand the vertical extent. Despite the vertical flipping, the amplitude remains 1 because:
  • The vertical distance from the midline to a peak is still 1, though reflected.
  • This reflection leads to peaks at \(-1\) and troughs at \(1\), which are the reverse of \(\sin x\).
  • Orientation doesn't affect the amplitude as it's a measure of distance.
Recognizing amplitude's consistency aids in clearly plotting peaks and troughs, creating an accurate representation of the sine function.

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Most popular questions from this chapter

Electric Circuit After the switch is closed in the circuit shown, the current \(t\) seconds later is \(I(t)=0.8 e^{-3 t} \sin 10 t\) . Find the current at the times (a) \(t=0.1 \mathrm{s}\) and (b) \(t=0.5 \mathrm{s}\)

Sound Vibrations \(\quad\) A tuning fork is struck, producing a pure tone as its tines vibrate. The vibrations are modeled by the function $$ v(t)=0.7 \sin (880 \pi t) $$ where \(v(t)\) is the displacement of the tines in millimeters at time \(t\) seconds. (a) Find the period of the vibration. (b) Find the frequency of the vibration, that is, the number of times the fork vibrates per second. (c) Graph the function \(v\) .

Find the exact value of the trigonometric function at the given real number. $$ \sin \left(-\frac{3 \pi}{2}\right) \quad \text { (b) } \cos \left(-\frac{3 \pi}{2}\right) \quad \text { (c) } \cot \left(-\frac{3 \pi}{2}\right) $$

Find the maximum and minimum values of the function. $$ y=2 \sin x+\sin ^{2} x $$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{3}{5}, \frac{4}{5}\right) $$
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