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.
e. Stretch vertically by a factor of 5
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 54a
Describe how each graph is obtained from the graph of π = Ζ(x).
a. π = Ζ(x - 5)
Verified step by step guidance1
Start with the graph of the function π = Ζ(x). This is your base graph.
Identify the transformation applied to the function. In this case, the transformation is π = Ζ(x - 5).
Recognize that the expression (x - 5) inside the function indicates a horizontal shift. Specifically, it is a shift to the right.
To apply this transformation, take each point (x, y) on the original graph of π = Ζ(x) and move it 5 units to the right. This means you will replace each x-coordinate with (x + 5).
Plot the new points to obtain the graph of π = Ζ(x - 5). The shape of the graph remains unchanged, but its position is shifted 5 units to the right.
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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 shifts, stretches, or reflections. These transformations can change the position and shape of the graph without altering its fundamental properties. Understanding how to manipulate the function's equation allows one to predict how the graph will change.
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5:25
Intro to Transformations
Horizontal Shifts
A horizontal shift occurs when the graph of a function is moved left or right along the x-axis. For the function πΆ = Ζ(x - 5), the graph is shifted 5 units to the right. This transformation is achieved by replacing x with (x - c), where c is the number of units shifted, affecting the input values of the function.
Recommended video:
5:25Intro to Transformations
Graphing Functions
Graphing functions involves plotting points that represent the output of a function for various input values. Understanding how to graph functions is essential for visualizing transformations. By knowing the original graph of πΆ = Ζ(x), one can easily apply transformations to see how the graph changes, enhancing comprehension of function behavior.
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5:53Graph of Sine and Cosine Function
Related Practice
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).
e. π = Ζ( x ) - 4
3
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
Textbook Question
For Exercises 51β54, solve for the angle ΞΈ, where 0 β€ ΞΈ β€ 2Ο.
cos 2ΞΈ + cos ΞΈ = 0
1
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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)