Textbook Question
General Sine Curves
For
f(x) = A sin ((2Ο/B)(x β C) +D
identify A, B, C, and D for the sine functions in Exercises 67β70 and sketch their graphs.
y = L/2Ο sin (2Οt/L), L > 0
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.27
In Exercises 19β32, find the (a) domain and (b) range.
π = cos(x - 3) + 1
Verified step by step guidance1
Step 1: Understand the function y = cos(x - 3) + 1. This is a transformation of the basic cosine function, where the graph is shifted horizontally by 3 units to the right and vertically by 1 unit upwards.
Step 2: Determine the domain of the function. The cosine function, cos(x), is defined for all real numbers. Therefore, the domain of y = cos(x - 3) + 1 is also all real numbers, which can be expressed as (-β, β).
Step 3: Analyze the range of the function. The basic cosine function, cos(x), has a range of [-1, 1]. The transformation y = cos(x - 3) + 1 shifts the entire range up by 1 unit.
Step 4: Calculate the new range. By shifting the range [-1, 1] up by 1 unit, the new range becomes [0, 2]. This is because the minimum value -1 becomes 0 and the maximum value 1 becomes 2.
Step 5: Summarize the findings. The domain of the function y = cos(x - 3) + 1 is (-β, β), and the range is [0, 2].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function y = cos(x - 3) + 1, the cosine function is defined for all real numbers, so the domain is all real numbers, denoted as (-β, β).
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Range of a Function
The range of a function is the set of all possible output values (y-values) that the function can produce. For y = cos(x - 3) + 1, the cosine function oscillates between -1 and 1, so when shifted up by 1, the range becomes [0, 2].
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Transformation of Functions
Transformations involve shifting, stretching, or compressing the graph of a function. In y = cos(x - 3) + 1, the graph of cos(x) is horizontally shifted right by 3 units and vertically shifted up by 1 unit, affecting the range but not the domain.
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Related Practice
Textbook Question
In Exercises 9β16, determine whether the function is even, odd, or neither.
π = xβ΄ + 1
xΒ³ - 2x
1
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Textbook Question
Graph the functions in Exercises 37β56.
y = (x + 2)Β³/Β² + 1
1
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Textbook Question
Graph the functions in Exercises 37β56.
y = (1/xΒ²) β 1
2
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Textbook Question
Graph the functions in Exercises 13β22. What is the period of each function?
βcos 2Οx
Textbook Question
A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs \(5/ft for the legs and \)10/ft for the hypotenuse, write the total cost C of construction as a function of h.