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Chapter 5: Problem 9

Graph the function. $$f(x)=-2+\sin x$$

Short Answer

Expert verified
The function \(f(x) = -2 + \sin x\) is a sine wave shifted 2 units down.

Step by step solution

01

Identify the Basic Function

The given function is related to the basic sine function, which is \(f(x) = \sin x\). This function is periodic, with a period of \(2\pi\), a maximum value of 1, and a minimum value of -1.
02

Analyze the Transformation

The function given is \(f(x) = -2 + \sin x\). This modifies the basic sine function by vertically shifting it down by 2 units. This means that every point on \(\sin x\) is moved downward by 2 on the y-axis.
03

Determine Key Points to Plot

Identify key points from the basic sine function and apply the transformation. For example, the point \((0, 0)\) on \(\sin x\) becomes \((0, -2)\), the point \((\pi/2, 1)\) becomes \((\pi/2, -1)\), and the point \((\pi, 0)\) becomes \((\pi, -2)\).
04

Graph the Transformed Function

Plot the transformed key points: \((0, -2)\), \((\pi/2, -1)\), \((\pi, -2)\), \((3\pi/2, -3)\), and \((2\pi, -2)\). Connect these points smoothly, following the sinusoidal wave pattern, ensuring the function maintains its periodic nature.
05

Check the Graph for Accuracy

Ensure that the graph reflects the correct period and amplitude. The function should have an amplitude of 1 (the same as \(\sin x\)), a period of \(2\pi\), a maximum of -1, and a minimum of -3 due to the vertical shift.

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Key Concepts

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

Understanding the Sine Function
The sine function is one of the fundamental trigonometric functions often used to model wave-like phenomena. It is represented as \( f(x) = \sin x \). The graph of the sine function is distinct due to its wave-like, smooth oscillations. Here are some core characteristics of the sine wave:
  • The period of the sine function is \( 2\pi \). This means that the function completes one full cycle every \( 2\pi \) units along the x-axis.
  • The sine wave oscillates between a maximum value of 1 and a minimum value of -1.
  • The wave crosses the x-axis at regular intervals: at points \((0, 0)\), \((\pi, 0)\), and \((2\pi, 0)\), and so on.
  • Its amplitude, which is half the distance between its maximum and minimum values, is 1.
The sine function is continuous and smooth, making it ideal for graphing and transformation exercises.
Exploring Periodic Functions
Periodic functions have a characteristic property of repeating their values in regular intervals or periods. The sine function is a prime example, as it repeats every \(2\pi\) units. This periodic nature is crucial for analyzing oscillating systems like sound and light waves. Here are some insights into periodic functions:
  • A function is considered periodic if it satisfies \( f(x+P) = f(x) \) for all values of \( x \), where \( P \) is the period.
  • The period is the length of the smallest interval over which the function repeats itself.
  • Periodic functions provide insight into cyclic phenomena that appear in nature, such as seasonal changes and biological rhythms.
  • Graphing a periodic function like \( \sin x \) involves recognizing its repeated pattern and symmetry.
Understanding the periodic nature of functions lets one predict future values and model natural occurrences.
The Role of Vertical Shifts
Vertical shifts are a type of transformation applied to functions to move their graphs up or down the y-axis. This concept is vital in modifying the position of the sine wave without altering its period or amplitude. Let's look at vertical shifts in more detail:
  • A vertical shift occurs when you add or subtract a constant from the function. For example, in \( f(x) = -2 + \sin x \), the "+\(-2\)" indicates a vertical downward shift by 2 units.
  • Despite the shift, the sine wave maintains its period of \(2\pi\) and amplitude of 1.
  • The minimum and maximum values are adjusted according to the shift: the maximum becomes -1, and the minimum shifts to -3.
  • Vertical shifts are often used to align a function with certain conditions or datasets by adjusting its vertical position.
Learning to apply vertical shifts enables flexibility in graphing functions to represent different scenarios or conditions effectively.

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Most popular questions from this chapter

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t\) \(\cos t,\) and \(\tan t.\) $$\left(-\frac{1}{3}, \frac{2 \sqrt{2}}{3}\right)$$

For each sine curve find the amplitude, period, phase, and horizontal shift. $$y=8 \sin 4\left(t+\frac{\pi}{12}\right)$$

The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is \(10^{\circ} .\) We know from physical principles that the angle \(\theta\) between the pendulum and its rest position changes in simple harmonic fashion. Find an equation that describes the size of the angle \(\theta\) as a function of time. (Take \(t=0\) to be a time when the pendulum is vertical.) (Figure cant copy)

For each sine curve find the amplitude, period, phase, and horizontal shift. $$y=5 \sin \left(2 t-\frac{\pi}{2}\right)$$

When a car with its horn blowing drives by an observer, the pitch of the horn seems higher as it approaches and lower as it recedes (see the figure below). This phenomenon is called the Doppler effect. If the sound source is moving at speed \(v\) relative to the observer and if the speed of sound is \(v_{0}\), then the perceived frequency \(f\) is related to the actual frequency \(f_{0}\) as follows. $$f=f_{0}\left(\frac{v_{0}}{v_{0} \pm v}\right)$$ We choose the minus sign if the source is moving toward the observer and the plus sign if it is moving away. Suppose that a car drives at \(110 \mathrm{ft} / \mathrm{s}\) past a woman standing on the shoulder of a highway, blowing its horn, which has a frequency of \(500 \mathrm{Hz}\). Assume that the speed of sound is \(1130 \mathrm{ft} / \mathrm{s} .\) (This is the speed in dry air at \(70^{\circ} \mathrm{F}\).) (a) What are the frequencies of the sounds that the woman hears as the car approaches her and as it moves away from her? (b) Let \(A\) be the amplitude of the sound. Find functions of the form $$y=A \sin \omega t$$ that model the perceived sound as the car approaches the woman and as it recedes. (Figure cant copy)
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