교과서 질문
Find the average rate of change of f(x) = x^2 - 4x from x_1 = 5 to x_2 = 9.
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
3장, 문제 55
In Exercises 55–59, use the graph of to graph each function g.
g(x) = f(x + 2) + 3
검증된 단계별 안내1
Identify the base function f(x) from the graph. Here, f(x) is an absolute value function with vertex at (0,0), forming a V-shape.
Understand the transformation in g(x) = f(x + 2) + 3. The term (x + 2) inside the function indicates a horizontal shift, and the +3 outside indicates a vertical shift.
Apply the horizontal shift: Replace x with (x + 2) shifts the graph of f(x) to the left by 2 units. This moves the vertex from (0,0) to (-2,0).
Apply the vertical shift: Adding +3 outside the function shifts the graph up by 3 units. This moves the vertex from (-2,0) to (-2,3).
Sketch the new graph g(x) by shifting every point of f(x) left 2 units and up 3 units, maintaining the same shape and slopes of the V.
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.
Function Transformations
Function transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this problem, the function g(x) = f(x + 2) + 3 represents a horizontal shift left by 2 units and a vertical shift up by 3 units of the original function f(x). Understanding these shifts helps in graphing the transformed function accurately.
추천 영상:
4:22
Domain & Range of Transformed Functions
Horizontal Shifts
A horizontal shift occurs when the input variable x is replaced by (x + h) or (x - h). For g(x) = f(x + 2), the graph of f(x) shifts left by 2 units because adding inside the function moves the graph in the opposite direction of the sign. This concept is crucial for repositioning the vertex of the V-shaped graph.
추천 영상:
5:34Shifts of Functions
Vertical Shifts
A vertical shift happens when a constant is added or subtracted outside the function, such as f(x) + k. In g(x) = f(x + 2) + 3, adding 3 shifts the entire graph of f(x + 2) upward by 3 units. This moves the vertex and all points on the graph vertically, maintaining the shape but changing the position.
추천 영상:
5:34Shifts of Functions
관련 실천
교과서 질문
In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. (go f) (-1)
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교과서 질문
Use the vertical line test to identify graphs in which y is a function of x.
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교과서 질문
Find
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = x+4, g(x) = 2x + 1
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교과서 질문
Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² – 10x – 6y – 30 = 0
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교과서 질문
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g (x) = √(x + 2) (x = = −2, −1, 2, 7)
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