Textbook Question
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
4 - x² |4 - x²|
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 53e
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
Verified step by step guidance1
Start with the original function g(x). To stretch the graph vertically, you will multiply the entire function by a constant factor.
Identify the factor by which you want to stretch the graph vertically. In this case, the factor is 5.
Multiply the function g(x) by the factor 5. This means you will write the new function as 5 * g(x).
Understand that multiplying the function by 5 will make all the y-values of the graph five times larger, effectively stretching the graph vertically.
The equation for the vertically stretched graph is y = 5 * g(x). This represents the transformation applied to the original graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Stretch
A vertical stretch occurs when the output values of a function are multiplied by a factor greater than 1. For example, if the function g(x) is transformed to 5g(x), every point on the graph of g is moved away from the x-axis by a factor of 5, effectively stretching the graph vertically. This transformation increases the height of the graph while maintaining the same x-coordinates.
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5:25
Intro to Transformations
Function Transformation
Function transformations involve altering the graph of a function through shifts, stretches, or reflections. These transformations can be represented mathematically by modifying the function's equation. Understanding how to apply these transformations allows one to predict the new position and shape of the graph based on the original function.
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5:25Intro to Transformations
Graph of a Function
The graph of a function visually represents the relationship between the input (x-values) and output (y-values) of the function. Each point on the graph corresponds to a pair (x, g(x)). Analyzing the graph helps in understanding the behavior of the function, including its intercepts, slopes, and overall shape, which are crucial when applying transformations like stretching or shifting.
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5:53Graph of Sine and Cosine Function
Related Practice
Textbook Question
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
__ ___
√ x √ |x|
Textbook Question
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
x³ |x³|
Textbook Question
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
a. 𝔂 = ƒ(x - 5)
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 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
cos 2θ + cos θ = 0
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