교과서 질문
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|
225
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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 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
1장, 문제 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
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
5:53Graph of Sine and Cosine Function
관련 실천
교과서 질문
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|
220
views
교과서 질문
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|
204
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교과서 질문
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
a. 𝔂 = ƒ(x - 5)
254
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교과서 질문
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
355
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교과서 질문
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
cos 2θ + cos θ = 0
220
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