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Chapter 1: Problem 27

\(f(x)=-2 \sin x\) \(g(x)=4 \sin x\)

Short Answer

Expert verified
The functions \(f(x) = -2 \sin x\) and \(g(x) = 4 \sin x\) are both sinusoidal functions. Compared to \(f(x)\), \(g(x)\) has a higher amplitude, meaning it covers a larger range above and below the x-axis. Also, \(f(x)\) unlike \(g(x)\), is a reflection of the standard sine graph over the x-axis, so it starts at zero and moves downwards instead of upwards.

Step by step solution

01

Identify the amplitude and direction of each function

The standard form of a sinusoidal function is \(a \sin (b x + c) + d\), where \(a\) signifies the amplitude of the function. Looking at the given functions, you can see that the amplitude of \(f(x)\) is 2 and the amplitude of \(g(x)\) is 4. Additionally, the negative sign in \(f(x)\) specifies that it is a reflection in the x-axis compared with the usual sine graph.
02

Compare the amplitudes

Amplitude pertains to the maximum reach or distance of the wave from its equilibrium position. For \(f(x)\), the amplitude is 2, which means it reaches a maximum of 2 units above and below the x-axis. For \(g(x)\), the amplitude is 4, indicating it reaches up to 4 units above and below the x-axis. Therefore, \(g(x)\) has a larger amplitude than \(f(x)\).
03

Compare the directions

The negative sign in \(f(x)=-2 \sin x\) means it is a reflection of the standard sine graph around the x-axis. So, while the standard sine graph and \(g(x)\) begin from zero and move upwards, \(f(x)\) begins from zero and moves downwards.

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Key Concepts

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

Amplitude
Amplitude is a key element in understanding trigonometric functions. When we talk about amplitude, we mean the height of the wave. This is determined by how far the wave reaches from its central or equilibrium line, which is often the x-axis.
For the function \( f(x) = -2 \sin x \), the amplitude is 2. This means that the wave peaks at 2 units above and dips 2 units below the central line. Similarly, for \( g(x) = 4 \sin x \), the amplitude is 4. This indicates larger waves, reaching up to 4 units above and below the x-axis.
  • A higher amplitude suggests taller waves, indicating more pronounced peaks and troughs.
  • The amplitude value is always positive and describes the height from the central line regardless of direction.
Recognizing the amplitude helps us in sketching or interpreting the graph of sine functions effectively.
Sine Function
The sine function is one of the fundamental trigonometric functions you'll encounter. It is periodic and oscillates between its peaks and valleys, repeating its pattern over specific intervals. This typical wave-like pattern is highly useful in various scientific and mathematical applications.
For \( f(x) = -2 \sin x \) and \( g(x) = 4 \sin x \), the sine function essentially acts as a wave generator; its form is key to understanding these oscillations.
  • In the sine function, \( \sin x \), the wave starts at 0, reaches a maximum, descends through 0 to a minimum, and returns back to 0.
  • The interval it takes to complete one full cycle is called the period, which is \(2\pi\) for the standard sine function.
  • Both functions \( f(x)\) and \( g(x)\) retain this periodicity, but due to their different amplitudes, their waves will look different in size.
Understanding the sine function's behavior allows us to explore how it models repetitive phenomena in the real world.
Reflection
Reflection in a mathematical context often deals with flipping a shape across a specific axis. For trigonometric functions, reflection changes the direction of the wave compared to the standard function.
In the function \( f(x) = -2 \sin x \), the negative symbol reflects the graph across the x-axis. It essentially flips the usual rise and fall of the sine wave.
  • This means that while a standard \( \sin x \) graph starts from 0 and moves upwards, \( f(x) = -2 \sin x \) begins by going downwards.
  • This reflection doesn't affect the amplitude magnitude, only the direction, making this a critical aspect in graph transformations.
  • The function \( g(x) = 4 \sin x \), on the other hand, follows the standard pattern without reflection, rising upwards initially.
Reflection helps in visualizing transformations and understanding how changes in an equation affect the graph's orientation.

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Most popular questions from this chapter

In Exercises 83-88, use a graphing utility to graph the function. $$ f(x)=\pi-\sin ^{-1}\left(\frac{2}{3}\right) $$

Write an algebraic expression that is equivalent to the expression. $$ \sec [\arcsin (x-1)] $$

In Exercises 109-112, sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \cos \theta=\frac{5}{6} $$

In Exercises 43-48, use the properties of inverse trigonometric functions to evaluate the expression. $$ \tan (\arctan 25) $$

In Exercises 69 and 70, use a graphing utility to graph \(f\) and \(g\) in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the graphs. $$ f(x)=\sin (\arctan 2 x), \quad g(x)=\frac{2 x}{\sqrt{1+4 x^{2}}} $$
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