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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 56d
Capitolo 1, Problema 56d

Use shifts and scalings to transform the graph of ƒ(x)=xƒ(x)=\(\sqrt{x}\)  into the graph of g. Use a graphing utility to check your work.
g(x)=3x−1−5g(x)=3\(\sqrt{x-1}\)-5

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Start with the base function \( f(x) = \sqrt{x} \).
Apply a horizontal shift to the right by 1 unit to get \( \sqrt{x-1} \). This accounts for the \( x-1 \) inside the square root.
Apply a vertical stretch by a factor of 3 to get \( 3\sqrt{x-1} \). This is done by multiplying the entire function by 3.
Apply a vertical shift downward by 5 units to get \( 3\sqrt{x-1} - 5 \). This is done by subtracting 5 from the entire function.
The transformed function is \( g(x) = 3\sqrt{x-1} - 5 \). Use a graphing utility to verify the transformation visually.

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Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant to the input (x) shifts the graph horizontally, while adding a constant to the output (f(x)) shifts it vertically. Understanding these transformations is crucial for manipulating the graph of a function like f(x) = √x to achieve the desired form of g(x).
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Intro to Transformations

Vertical and Horizontal Shifts

Vertical shifts occur when a constant is added or subtracted from the function, affecting its position along the y-axis. Horizontal shifts are achieved by adding or subtracting a constant from the input variable, affecting the x-axis position. In the transformation from f(x) = √x to g(x) = 3√(x-1) - 5, the term (x-1) indicates a horizontal shift to the right by 1 unit, while the -5 indicates a vertical shift downward by 5 units.
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Intro to Transformations

Scaling and Stretching

Scaling refers to multiplying the function by a constant, which affects the steepness of the graph. A vertical scaling, such as multiplying by 3 in g(x) = 3√(x-1) - 5, stretches the graph away from the x-axis, making it steeper. Understanding how scaling impacts the graph is essential for accurately transforming f(x) into g(x) and predicting the resulting shape of the graph.
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Intro to Transformations
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