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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 21
Chapter 3, Problem 21

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

g(x) = f(x-1)+2

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1
Identify the given function transformation: \(g(x) = f(x-1) + 2\). This means the graph of \(f(x)\) is shifted horizontally and vertically.
Understand the horizontal shift: The term \((x-1)\) inside the function indicates a shift to the right by 1 unit. So, every point \((x, y)\) on \(f(x)\) moves to \((x+1, y)\) on \(g(x)\).
Understand the vertical shift: The \(+2\) outside the function means the graph is shifted up by 2 units. So, every point \((x, y)\) on \(f(x)\) moves to \((x, y+2)\) on \(g(x)\).
Apply both transformations to each key point on the graph of \(f(x)\): For example, the point \((-2, 0)\) on \(f(x)\) will move to \((-2+1, 0+2) = (-1, 2)\) on \(g(x)\). Repeat this for the points \((0, -4)\) and \((2, 0)\).
Plot the new points on the coordinate plane and connect them with the same shape as the original graph to complete 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, stretching, compressing, or reflecting the graph of a function. In this problem, the function g(x) = f(x - 1) + 2 represents a horizontal shift to the right by 1 unit and a vertical shift upward by 2 units of the original function f(x). Understanding these shifts helps in accurately graphing the transformed function.
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Domain & Range of Transformed Functions

Horizontal Shifts

A horizontal shift occurs when the input variable x is replaced by (x - h), shifting the graph h units to the right if h is positive, or to the left if h is negative. For g(x) = f(x - 1), the graph of f(x) moves 1 unit to the right. This affects the x-coordinates of all points on the graph.
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Shifts of Functions

Vertical Shifts

A vertical shift happens when a constant k is added to the function, changing it to f(x) + k. This moves the graph up by k units if k is positive, or down if k is negative. In g(x) = f(x - 1) + 2, adding 2 shifts the entire graph of f(x - 1) upward by 2 units, affecting the y-coordinates.
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Shifts of Functions
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