Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 52

Chapter 3, Problem 52

Use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)

Verified step by step guidance

Step 1: Understand the given function transformation. The function g(x) = 2f(x-1) involves two transformations applied to the base function f(x): a horizontal shift and a vertical stretch.

Step 2: Identify the horizontal shift. The term (x-1) inside the function indicates a horizontal shift to the right by 1 unit. This means that every point on the graph of f(x) will move 1 unit to the right.

Step 3: Identify the vertical stretch. The coefficient 2 outside the function indicates a vertical stretch by a factor of 2. This means that the y-values of the graph of f(x) will be multiplied by 2, making the graph taller.

Step 4: Apply the transformations step by step. First, shift the graph of f(x) 1 unit to the right. Then, multiply the y-values of the shifted graph by 2 to apply the vertical stretch.

Step 5: Sketch the resulting graph of g(x). Use the transformed points to plot the new graph, ensuring that the horizontal shift and vertical stretch are accurately represented.

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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 changes made to the graph of a function based on modifications to its equation. Common transformations include vertical and horizontal shifts, stretches, and reflections. In the case of g(x) = 2f(x-1), the function is horizontally shifted to the right by 1 unit and vertically stretched by a factor of 2.

Horizontal Shift

A horizontal shift occurs when the input of a function is altered, resulting in the graph moving left or right. For g(x) = 2f(x-1), the 'x-1' indicates a shift to the right by 1 unit. This means that every point on the graph of f(x) will be moved to the right, affecting the x-coordinates of the graph.

Vertical Stretch

A vertical stretch occurs when the output of a function is multiplied by a factor greater than 1, which increases the distance of points from the x-axis. In g(x) = 2f(x-1), the factor of 2 indicates that the graph of f(x) will be stretched vertically, making it taller. This transformation affects the y-coordinates of the graph, amplifying the values of f(x) by a factor of 2.

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Stretches & Shrinks of Functions

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