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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 99
Chapter 1, Problem 99

Identify the amplitude and period of the following functions.
q(x)=3.6cos(πx24)q\(\left\)(x\(\right\))=3.6\(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\))

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The function given is q(x) = 3.6 \(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\)). This is a cosine function of the form a \(\cos\)(bx), where 'a' is the amplitude and 'b' affects the period.
Identify the amplitude: In the function q(x) = 3.6 \(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\)), the coefficient 'a' in front of the cosine function is 3.6. Therefore, the amplitude is 3.6.
Determine the period: The period of a cosine function a \(\cos\)(bx) is given by \(\frac{2\pi}{b}\).
In the function q(x) = 3.6 \(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\)), the value of 'b' is \(\frac{\pi}{24}\).
Calculate the period using the formula: Substitute b = \(\frac{\pi}{24}\) into the period formula \(\frac{2\pi}{b}\) to find the period of the function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central position, which in the context of a cosine function is the coefficient in front of the cosine term. For the function q(x) = 3.6cos(πx/24), the amplitude is 3.6, indicating that the function oscillates between 3.6 and -3.6.
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Period

The period of a function describes the length of one complete cycle of the wave. For a cosine function of the form cos(kx), the period can be calculated using the formula P = 2π/k. In this case, k is π/24, leading to a period of P = 2π/(π/24) = 48, meaning the function completes one full cycle every 48 units along the x-axis.
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Cosine Function

The cosine function is a fundamental trigonometric function that describes the relationship between the angle and the lengths of the sides of a right triangle. It is periodic, oscillating between -1 and 1, and is often used to model wave-like phenomena. In the given function, the cosine term dictates the shape and behavior of the wave, influenced by its amplitude and period.
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Related Practice
Textbook Question

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle θ\(\theta\) (measured in radians) is A=12r2θA=\(\frac\)12r^2\(\theta\).

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Textbook Question

{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

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Textbook Question

Identify the amplitude and period of the following functions.

f(π)=2sin2θf\(\left\)(\(\pi\]\right\))=2\(\sin\)2\(\theta\)

Textbook Question

Beginning with the graphs of y=sinxy=\(\sin\) x or y=cosxy=\(\cos\) x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

q(x)=3.6cos(πx24)+2q\(\left\)(x\(\right\))=3.6\(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\))+2

Textbook Question

Identify the amplitude and period of the following functions.

g(θ)=3cos(θ3)g\(\left\)(\(\theta\]\right\))=3\(\cos\[\left\)(\(\frac{\theta}{3}\]\right\))