Textbook Question
Derive a formula for tan (A β B).
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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 54d
Describe how each graph is obtained from the graph of π = Ζ(x).
d. π = Ζ(2x + 1)
Verified step by step guidance1
Start with the graph of π = Ζ(x). This is your base graph, which represents the function Ζ(x) plotted on the coordinate plane.
Consider the expression inside the function: 2x + 1. This indicates a transformation of the graph of Ζ(x).
The term '2x' suggests a horizontal compression. Since the factor is greater than 1, the graph will be compressed horizontally by a factor of 1/2. This means that each x-coordinate of the original graph will be halved.
Next, the '+1' inside the function indicates a horizontal shift. Specifically, it shifts the graph to the left by 1 unit. This is because the transformation is inside the function, affecting the x-values directly.
Combine these transformations: first, compress the graph horizontally by a factor of 1/2, and then shift it 1 unit to the left. The resulting graph is the graph of π = Ζ(2x + 1).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the process of altering the graph of a function through various operations, such as shifting, stretching, or compressing. These transformations can be vertical or horizontal, affecting the position and shape of the graph. Understanding how these transformations work is essential for predicting how the graph of a function will change when its equation is modified.
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5:25
Intro to Transformations
Horizontal Shifts
A horizontal shift occurs when the input variable of a function is altered, resulting in the graph moving left or right. In the equation πΆ = Ζ(2x + 1), the term '2x + 1' indicates a horizontal shift. Specifically, the graph shifts to the left by 0.5 units, as the transformation involves solving for x in the form of 'x = (y - 1)/2'.
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5:25Intro to Transformations
Horizontal Scaling
Horizontal scaling involves stretching or compressing the graph of a function along the x-axis. In the equation πΆ = Ζ(2x + 1), the coefficient '2' in front of x indicates a horizontal compression by a factor of 2. This means that for every unit increase in the output, the input must increase by only half a unit, effectively making the graph narrower.
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04:50Critical Points
Related Practice
Textbook Question
Describe how each graph is obtained from the graph of π = Ζ(x).
e. π = Ζ( x ) - 4
3
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Textbook Question
Describe how each graph is obtained from the graph of π = Ζ(x).
a. π = Ζ(x - 5)
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Textbook Question
Shifting and Scaling Graphs
Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.
f. Compress horizontally by a factor of 5
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Textbook Question
For Exercises 51β54, solve for the angle ΞΈ, where 0 β€ ΞΈ β€ 2Ο.
cos 2ΞΈ + cos ΞΈ = 0
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Textbook Question
In Exercises 55β58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15β1.17, and applying an appropriate transformation.
y = - β(1 + x/2)
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