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Chapter 0: Problem 28

True-False Determine whether the statement is true or false. Explain your answer. Curves in the family \(y=-5 \sin (A \pi x)\) have amplitude 5 and period \(2 /|A|\).

Short Answer

Expert verified
The statement is true; the curves have amplitude 5 and period \( \frac{2}{|A|} \).

Step by step solution

01

Understand Amplitude

The amplitude of a sine function \(y = -5 \sin(A \pi x)\) is the absolute value of the coefficient in front of the sine. The function here is \(-5 \), which means the amplitude is \(|-5| = 5\). This part of the statement about amplitude is true.
02

Derive Period Formula of Sine Function

The standard form for the period of a function \(\sin(Bx)\) is \(\frac{2\pi}{|B|}\). For \(\sin(A\pi x)\), replace \(B\) with \(A\pi\), so the period is \(\frac{2\pi}{|A\pi|} \), which simplifies to \(\frac{2}{|A|}\). This matches the period given in the statement.
03

Verify Overall Statement

Now that we have verified both the amplitude and the period according to the given formula for the curve \(y = -5 \sin(A \pi x)\), we can affirm that both conditions in the statement are true.

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Key Concepts

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

amplitude
The amplitude of a sine function refers to its height from the origin (middle point) to its maximum or minimum value. In simpler terms, it is the "stretch" of the wave vertically. For a sine function like \(y = -5 \sin(A \pi x)\), the amplitude is obtained by taking the absolute value of the coefficient in front of the sine function.

In our case, this coefficient is \(-5\). The absolute value helps us ignore the negative sign because amplitude is always a positive quantity. So,
  • Amplitude = \(|-5| = 5\)
This means that the curve will oscillate 5 units above and below the center line, making this part of the original statement true.
period of sine function
The period of a sine function determines how long it takes for the wave to complete one full cycle. Imagine a single cycle as a movement through a crest, a trough, and back to the starting point. For the general sine function \(\sin(Bx)\), the period is calculated using the formula
  • Period = \(\frac{2\pi}{|B|}\)
Now, in the function \(y = -5 \sin(A \pi x)\), compare it with \(\sin(Bx)\) and you'll realize \(B = A\pi\). This means the calculation for period will adjust similarly:
  • Period = \(\frac{2\pi}{|A\pi|} = \frac{2}{|A|}\)

This expression matches the original statement describing the period, confirming its accuracy.
sine function properties
Understanding the properties of sine functions is essential because they reveal consistent behaviors of these mathematical waves. Key properties include amplitude and period, which we have mentioned, but there's more.

Some notable properties include:
  • The sine function is an odd function, which means \(\sin(-x) = -\sin(x)\). This is evident in our equation because of the negative sign in front of the main term.
  • The graph of the sine function is a smooth, continuous wave that repeats itself at regular intervals.
  • It crosses the origin and has a maximum at \(\frac{\pi}{2}\), with minimum at \(-\frac{\pi}{2}\).

The nature of these properties helps in predicting and understanding the behavior of sine curves in equations, making them crucial tools in trigonometry.

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Most popular questions from this chapter

A variable \(y\) is said to be imversely proportional to the square of a variable \(x\) if \(y\) is related to \(x\) by an equation of the form \(y=k / x^{2}\), where \(k\) is a nonzero constant, called the constant of proportionality. This terminology is used in these exercises. According to Coulomb's law, the force \(F\) of attraction between positive and negative point charges is inversely proportional to the square of the distance \(x\) between them. (a) Assuming that the force of attraction between two point charges is \(0.0005\) newton when the distance between them is \(0.3\) meter, find the constant of proportionality (with proper units). (b) Find the force of attraction between the point charges when they are 3 meters apart. (c) Make a graph of force versus distance for the two charges. \(\quad(c a=r .)\) (d) What happens to the force as the particles get closer and closer together? What happens as they get farther and farther apart?

True-False Determine whether the statement is true or false. Explain your answer. The graph of an even function is symmetric about the \(y\) -axis.

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h .\) [Note: Each exercise has more than one solution.] (a) \(f(x)=\frac{1}{1-x^{2}}\) (b) \(f(x)=|5+2 x|\)

Use a calculating utility and the change of base formula (6) to find the values of \(\log _{2} 7.35\) and \(\log _{5} 0.6\), rounded to four decimal places.

In the United States, a standard electrical outlet supplies sinusoidal electrical current with a maximum voltage of \(V=120 \sqrt{2}\) volts \((\mathrm{V})\) at a frequency of 60 hertz \((\mathrm{Hz})\). Write an equation that expresses \(V\) as a function of the time \(t\), assuming that \(V=0\) if \(t=0 .\) [Note: \(1 \mathrm{~Hz}=1\) cycle per second.]
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