Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 76

Chapter 3, Problem 76

Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)

Verified step by step guidance

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. Its graph starts at the origin (0, 0) and curves upward to the right.

Next, analyze the given function, g(x) = 2√(x+1). Notice that this function is a transformation of the base function f(x). The transformations include a horizontal shift, a vertical stretch, and a vertical scaling.

The term (x+1) inside the square root indicates a horizontal shift. Specifically, the graph of f(x) = √x is shifted 1 unit to the left because of the +1 inside the square root.

The coefficient 2 outside the square root represents a vertical stretch. This means that the y-values of the graph are multiplied by 2, making the graph steeper.

To graph g(x), start by shifting the graph of f(x) = √x one unit to the left. Then, stretch the graph vertically by multiplying the y-values by 2. Plot key points such as (0, 0), (-1, 0), and others to visualize the transformation.

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Key Concepts

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

Square Root Function

The square root function, 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 x and its square root. Understanding this function is crucial as it serves as the foundation for applying transformations.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant inside the function's argument shifts the graph horizontally, while multiplying the function by a constant stretches or compresses it vertically. These transformations allow us to manipulate the basic shape of the square root function to create new functions like g(x) = 2√(x+1).

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of transformations that affect the position of a graph. A horizontal shift occurs when a constant is added or subtracted from the input variable (x), moving the graph left or right. A vertical shift occurs when a constant is added or subtracted from the output of the function, moving the graph up or down. In g(x) = 2√(x+1), the '+1' indicates a leftward shift of the graph by 1 unit.

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Related Practice

Textbook Question

Use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

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Textbook Question

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

Textbook Question

Use the graph of g to solve Exercises 71–76.

For what value of x is g(x) = 1?

Textbook Question

Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).

h(x) = (3x − 1)⁴

Textbook Question

Use the graph of g to solve Exercises 71–76.

For what value of x is g(x) = -1?

Textbook Question

In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

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