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Find the average rate of change of the function from x1 to x2. f(x) = 3x from x1 = 0 to x2 = 5
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 3, Problema 13
Use the graph of y = f(x) to graph each function g.
g(x) = f(x/2)
Guida verificata passo dopo passo1
Step 1: Understand the transformation g(x) = f(x/2). This transformation involves a horizontal stretch of the graph of f(x) by a factor of 2. Each x-coordinate of the points on the graph of f(x) will be multiplied by 2, while the y-coordinates remain unchanged.
Step 2: Identify the key points on the graph of f(x). From the graph, the points are (1, -3) and (4, -3). These points will be transformed according to the horizontal stretch.
Step 3: Apply the transformation to the x-coordinates of the points. Multiply each x-coordinate by 2: For (1, -3), the new x-coordinate is 1 * 2 = 2, so the transformed point is (2, -3). For (4, -3), the new x-coordinate is 4 * 2 = 8, so the transformed point is (8, -3).
Step 4: Plot the transformed points (2, -3) and (8, -3) on the graph. These points represent the new positions of the graph after the horizontal stretch.
Step 5: Draw the graph of g(x) = f(x/2) by connecting the transformed points with a horizontal line, maintaining the same y-value (-3) across the stretch.
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Function Transformation
Function transformation refers to the process of altering the graph of a function through various operations, such as shifting, stretching, or compressing. In this case, the function g(x) = f(x/2) represents a horizontal stretch of the original function f(x) by a factor of 2, meaning that every x-coordinate in the graph of f(x) is doubled in g(x).
Video consigliato:
4:22
Domain & Range of Transformed Functions
Horizontal Stretch
A horizontal stretch occurs when the x-values of a function are multiplied by a factor less than 1. For g(x) = f(x/2), the x-values are effectively halved, which stretches the graph horizontally. This transformation results in the graph of g(x) appearing wider than that of f(x), as points that were originally close together on f(x) will be spaced further apart on g(x).
Video consigliato:
6:02Stretches & Shrinks of Functions
Graph Interpretation
Graph interpretation involves analyzing the visual representation of a function to extract information about its behavior, such as intercepts, maxima, minima, and overall shape. In this exercise, understanding the graph of f(x) is crucial to accurately sketching g(x), as it allows one to see how the transformation affects the positions of key points and the overall structure of the graph.
Video consigliato:
02:16Graphs and Coordinates - Example
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