Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 82

Chapter 3, Problem 82

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

Verified step by step guidance

Start by graphing the parent function f(x) = |x|. This is the absolute value function, which forms a V-shaped graph with its vertex at the origin (0, 0). The graph opens upwards, with the left side having a slope of -1 and the right side having a slope of 1.

Understand the transformation applied to the parent function. The given function g(x) = |x| + 3 adds a constant value of 3 to the output of the absolute value function. This is a vertical shift.

Apply the vertical shift to the graph of f(x) = |x|. To do this, move every point on the graph of f(x) = |x| upward by 3 units. For example, the vertex (0, 0) of f(x) = |x| will move to (0, 3).

Adjust the slopes of the graph accordingly. The left side of the graph will still have a slope of -1, and the right side will still have a slope of 1, but the entire graph is now shifted upward.

Sketch the transformed graph of g(x) = |x| + 3. Label the vertex at (0, 3) and ensure the V-shape is preserved, with the left and right sides extending as described.

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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 opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial as it serves as the foundation for graphing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, adding a constant (like +3) to the absolute value function translates the graph vertically upwards by 3 units. Recognizing how these transformations affect the graph is essential for accurately sketching the new function.

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function's output. For g(x) = |x| + 3, the entire graph of f(x) = |x| is moved up by 3 units. This concept is important for understanding how the position of the graph changes without altering its shape.

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Related Practice

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Graph each linear function. 6x-5f(x) - 20 = 0

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

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Find (f+g)(−3).

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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.

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