Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problema 68

Capitolo 3, Problema 68

Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √x + 1

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Start by identifying the base function, f(x) = √x. This is the square root function, which has a domain of x ≥ 0 and a range of y ≥ 0. The graph of f(x) = √x starts at the origin (0, 0) and curves upward to the right.

Next, analyze the given function, g(x) = √x + 1. Notice that this function is a transformation of the base function f(x). Specifically, the '+1' outside the square root indicates a vertical shift upward by 1 unit.

To graph g(x), take the graph of f(x) = √x and shift every point on the graph upward by 1 unit. For example, the point (0, 0) on f(x) becomes (0, 1) on g(x), and the point (1, 1) on f(x) becomes (1, 2) on g(x).

Verify the transformation by substituting specific x-values into g(x). For example, when x = 0, g(0) = √0 + 1 = 1, and when x = 4, g(4) = √4 + 1 = 3. These points confirm the vertical shift.

Finally, sketch the graph of g(x) = √x + 1. It should look like the graph of f(x) = √x, but shifted upward by 1 unit. Label key points such as (0, 1), (1, 2), and (4, 3) to ensure accuracy.

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Square Root Function

The square root function, denoted as f(x) = √x, is defined for x ≥ 0 and produces non-negative outputs. Its graph is a curve that starts at the origin (0,0) and increases gradually, reflecting the relationship between the input x and its square root. Understanding this function is crucial as it serves as the foundation for applying transformations.

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Imaginary Roots with the Square Root Property

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the transformation applied to f(x) = √x to obtain g(x) = √x + 1 is a vertical shift upwards by 1 unit. Recognizing how these transformations affect the graph is essential for accurately sketching the new function.

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Intro to Transformations

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function's output. For g(x) = √x + 1, the '+1' indicates that every point on the graph of f(x) = √x is moved up by one unit. This concept is vital for understanding how the original graph is altered to create the new function's graph.

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Shifts of Functions

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