Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 92

Chapter 3, Problem 92

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = 2|x+3|

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Start by graphing the parent function f(x) = |x|. This is a V-shaped graph with its vertex at the origin (0, 0). The graph opens upwards, with a slope of 1 for x > 0 and a slope of -1 for x < 0.

Next, analyze the transformation inside the absolute value function. The term (x + 3) indicates a horizontal shift. Specifically, the graph of f(x) = |x| is shifted 3 units to the left. This means the vertex of the graph moves from (0, 0) to (-3, 0).

Now, consider the coefficient 2 outside the absolute value function. This represents a vertical stretch. The graph of f(x) = |x| is stretched by a factor of 2, meaning the slopes of the lines become steeper: 2 for x > -3 and -2 for x < -3.

Combine the transformations: Start with the vertex at (-3, 0), apply the vertical stretch, and ensure the slopes of the graph are adjusted accordingly. The left side of the graph will have a slope of -2, and the right side will have a slope of 2.

Finally, sketch the transformed graph h(x) = 2|x + 3| by applying the horizontal shift and vertical stretch to the parent function. Verify that the vertex is at (-3, 0) and the slopes are consistent with the transformations.

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Key Concepts

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

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function has a V-shaped graph that intersects the origin (0,0) and is symmetric about the y-axis. Understanding this function is crucial as it serves as the foundation for graphing transformations.

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the graph of a function. For the function h(x) = 2|x+3|, the graph of f(x) = |x| is first shifted left by 3 units and then vertically stretched by a factor of 2. Recognizing these transformations allows for accurate graphing of modified functions.

Graphing Techniques

Graphing techniques include plotting key points, identifying transformations, and understanding the behavior of functions. For h(x) = 2|x+3|, one must first graph the base function and then apply the transformations systematically. Mastery of these techniques is essential for visualizing and interpreting the behavior of complex functions.

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Textbook Question

Use the graphs of f and g to evaluate each composite function.

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Textbook Question

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

Use the graphs of f and g to evaluate each composite function.

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Textbook Question

The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^-1(x), the inverse function. (b) Verify that your equation is correct by showing that f(f^-1(x)) = x and f^-1(f(x)) = x. f(x) = 4x - 3

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Textbook Question

Let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g (1) and f(g(1)).

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