Textbook Question
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
sin (2π − x)
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.2.48
Graph the functions in Exercises 37–56.
y = (x + 2)³/² + 1
Verified step by step guidance1
Identify the basic form of the function: The given function is \( y = (x + 2)^{\frac{3}{2}} + 1 \). This is a transformation of the basic function \( y = x^{\frac{3}{2}} \).
Determine the transformations: The expression \( (x + 2) \) indicates a horizontal shift to the left by 2 units, and the \( +1 \) outside the power indicates a vertical shift upwards by 1 unit.
Consider the domain: Since the function involves \( (x + 2)^{\frac{3}{2}} \), the expression inside the power must be non-negative. Therefore, \( x + 2 \geq 0 \), which implies \( x \geq -2 \).
Sketch the basic function: Start by sketching the graph of \( y = x^{\frac{3}{2}} \), which is defined for \( x \geq 0 \) and resembles a square root function but grows faster.
Apply the transformations: Shift the graph of \( y = x^{\frac{3}{2}} \) 2 units to the left and 1 unit up to obtain the graph of \( y = (x + 2)^{\frac{3}{2}} + 1 \). Ensure the graph starts at the point \( (-2, 1) \) and follows the shape of the transformed function.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation involves shifting, stretching, or compressing the graph of a function. In the given function, y = (x + 2)^(3/2) + 1, the term (x + 2) indicates a horizontal shift to the left by 2 units, while the +1 indicates a vertical shift upwards by 1 unit. Understanding these transformations is crucial for accurately graphing the function.
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5:25
Intro to Transformations
Domain and Range
The domain of a function refers to the set of all possible input values (x-values), while the range refers to the set of all possible output values (y-values). For the function y = (x + 2)^(3/2) + 1, the domain is x ≥ -2, since the expression under the square root must be non-negative. The range starts from 1 and extends to infinity, as the minimum value occurs when x = -2.
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Graphing Techniques
Graphing techniques involve plotting points, identifying key features such as intercepts, and understanding the overall shape of the graph. For the function y = (x + 2)^(3/2) + 1, it is important to calculate specific points, such as the vertex and intercepts, and to recognize that the graph will have a characteristic shape due to the cubic root and the transformations applied.
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Related Practice
Textbook Question
General Sine Curves
For
f(x) = A sin ((2π/B)(x – C) +D
identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.
y = L/2π sin (2πt/L), L > 0
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x⁴ + 1
x³ - 2x
1
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = cos(x - 3) + 1
1
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Textbook Question
Graph the functions in Exercises 37–56.
y = (1/x²) − 1
2
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Textbook Question
Graph the functions in Exercises 13–22. What is the period of each function?
−cos 2πx