Textbook Question
Derive a formula for tan (A β B).
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 54e
Describe how each graph is obtained from the graph of π = Ζ(x).
e. π = Ζ( x ) - 4
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Verified step by step guidance1
Start with the graph of the function π = Ζ(x). This is your base graph from which transformations will be applied.
The expression π = Ζ(x) - 4 indicates a vertical shift. Specifically, subtracting 4 from the function means you will shift the entire graph of π = Ζ(x) downward by 4 units.
To visualize this, take each point (x, y) on the original graph of π = Ζ(x) and move it to the point (x, y - 4). This will lower every point on the graph by 4 units.
Ensure that the shape and orientation of the graph remain unchanged; only the vertical position is altered.
After applying the vertical shift, the new graph represents the function π = Ζ(x) - 4, which is the original graph moved down by 4 units.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Shifts
Vertical shifts occur when a constant is added to or subtracted from a function. In the case of πΆ = Ζ(x) - 4, the graph of Ζ(x) is shifted downward by 4 units. This transformation affects the y-coordinates of all points on the graph, while the x-coordinates remain unchanged.
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Intro to Transformations
Function Notation
Function notation, such as Ζ(x), represents a relationship where each input x corresponds to exactly one output y. Understanding this notation is crucial for interpreting how changes to the function, like subtracting a constant, affect the overall graph. It allows for clear communication of mathematical ideas and transformations.
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04:22Sigma Notation
Graph Transformations
Graph transformations refer to the various ways a function's graph can be altered, including shifts, stretches, and reflections. In this case, the transformation involves a vertical shift, which is a fundamental concept in understanding how the graph of a function can be manipulated without changing its shape.
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5:25Intro to Transformations
Related Practice
Textbook Question
Apply the formula for cos (A β B) to the identity sin ΞΈ = cos (Ο/2 β ΞΈ) to obtain the addition formula for sin (A + B).
Textbook Question
Describe how each graph is obtained from the graph of π = Ζ(x).
d. π = Ζ(2x + 1)
Textbook Question
Describe how each graph is obtained from the graph of π = Ζ(x).
a. π = Ζ(x - 5)
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)