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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 41
Chapter 3, Problem 41

Use the graph of y = f(x) to graph each function g. g(x) = -½ ƒ ( x + 2) - 2

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Start by understanding the transformations applied to the base function y = f(x). The given function g(x) = -½ ƒ(x + 2) - 2 involves multiple transformations: a horizontal shift, a vertical stretch/compression, a reflection, and a vertical shift.
Identify the horizontal shift: The term (x + 2) inside the function indicates a horizontal shift to the left by 2 units. This means every point on the graph of f(x) will move 2 units to the left.
Determine the vertical stretch/compression and reflection: The coefficient -½ in front of ƒ(x + 2) indicates two transformations. The negative sign reflects the graph across the x-axis, and the factor of ½ compresses the graph vertically by a factor of ½. This means the y-values of the graph will be halved and flipped in sign.
Account for the vertical shift: The -2 at the end of the function indicates a vertical shift downward by 2 units. This means every point on the graph will move 2 units down.
Combine all transformations: To graph g(x), start with the graph of y = f(x). First, shift the graph 2 units to the left. Then, reflect it across the x-axis and compress it vertically by a factor of ½. Finally, shift the resulting graph 2 units downward. Plot the transformed points 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 Transformation

Function transformation involves altering the graph of a function through various operations such as shifting, reflecting, stretching, or compressing. In the given function g(x) = -½ f(x + 2) - 2, the transformations include a horizontal shift to the left by 2 units, a vertical compression by a factor of ½, and a vertical shift downward by 2 units.
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Domain & Range of Transformed Functions

Horizontal Shifts

Horizontal shifts occur when the input of a function is adjusted by adding or subtracting a constant. For g(x) = -½ f(x + 2), the '+2' indicates a shift to the left by 2 units on the x-axis. This means that every point on the graph of f(x) will move leftward, affecting the overall position of the graph of g(x).
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Shifts of Functions

Vertical Shifts and Reflections

Vertical shifts involve moving the graph of a function up or down by adding or subtracting a constant from the function's output. In g(x), the '-2' indicates a downward shift of 2 units. Additionally, the negative sign in front of the ½ reflects the graph across the x-axis, inverting its orientation while also compressing it vertically by a factor of ½.
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Graphs of Shifted & Reflected Functions
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