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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 47
Chapter 3, Problem 47

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

Verified step by step guidance
1
Start with the original function \( y = f(x) \). The goal is to graph \( g(x) = -f(x-1) + 1 \) by applying transformations to \( f(x) \).
First, consider the inside of the function argument: \( f(x-1) \). This represents a horizontal shift of the graph of \( f(x) \) to the right by 1 unit.
Next, apply the negative sign in front of \( f(x-1) \), which means reflecting the graph of \( f(x-1) \) across the x-axis (flip it upside down).
Finally, add 1 to the entire function: \( -f(x-1) + 1 \). This shifts the graph vertically upward by 1 unit.
To summarize, start with the graph of \( f(x) \), shift it right by 1 unit, reflect it over the x-axis, then shift it up by 1 unit to obtain the graph of \( g(x) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Transformations

Function transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this problem, the function g(x) is derived from f(x) by applying horizontal shifts, reflections, and vertical shifts, which change the graph's position and orientation without altering its shape.
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Domain & Range of Transformed Functions

Horizontal Shifts

A horizontal shift moves the graph left or right. For g(x) = -f(x-1) + 1, the term (x-1) inside the function indicates a shift of the graph of f(x) one unit to the right. This means every point on f(x) moves right by 1 unit before other transformations are applied.
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Shifts of Functions

Reflections and Vertical Shifts

The negative sign before f(x-1) reflects the graph across the x-axis, flipping it upside down. The +1 outside the function shifts the entire graph up by one unit. Together, these transformations modify the graph's orientation and vertical position.
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Graphs of Shifted & Reflected Functions
Related Practice
Textbook Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x³, g(x) = x³ +2

Textbook Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 2)² + (y + 2)² = 4

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Textbook Question

Find ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

Textbook Question

In Exercises 46–49, give the slope and y-intercept of each line whose equation is given. Then graph the line. y = (2/5)x - 1

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Textbook Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x)= |x|, g(x) = |x| +1

Textbook Question

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = 9x/(x - 4), g(x) = 7/(x+8)

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