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 90

Capitolo 3, Problema 90

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -|x + 4| +2

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Start with the base graph of the absolute value function, f(x) = |x|. This graph is a V-shaped graph with its vertex at the origin (0, 0), opening upwards.

Identify the transformation inside the absolute value: x + 4. This represents a horizontal shift. Specifically, the graph of |x| is shifted 4 units to the left because of the +4 inside the absolute value.

Next, consider the negative sign in front of the absolute value, -|x + 4|. This reflects the graph of |x + 4| across the x-axis, causing the V-shape to open downwards instead of upwards.

Now, account for the +2 outside the absolute value. This represents a vertical shift. The entire graph of -|x + 4| is shifted 2 units upward.

Combine all the transformations: Start with the base graph of |x|, shift it 4 units to the left, reflect it across the x-axis, and finally shift it 2 units upward. Plot the resulting graph to visualize g(x) = -|x + 4| + 2.

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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 applying transformations to graph other functions.

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Function Composition

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, adding a constant inside the absolute value affects horizontal shifts, while adding outside affects vertical shifts. In the function g(x) = -|x + 4| + 2, the transformations include a horizontal shift left by 4 units, a reflection across the x-axis, and a vertical shift up by 2 units.

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Domain & Range of Transformed Functions

Graphing Techniques

Graphing techniques involve plotting points and understanding how transformations affect the shape and position of a graph. For the function g(x), one must first graph f(x) = |x|, then apply the identified transformations systematically. This process helps visualize the changes and accurately represent the new function on a coordinate plane.

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Graphs and Coordinates - Example

Pratica correlata

Domanda del libro di testo

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

(fog) (-1)

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Domanda del libro di testo

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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Domanda del libro di testo

Use the graphs of f and g to solve Exercises 83–90.

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Domanda del libro di testo

Use the graphs of f and g to solve Exercises 83–90.

Graph f+g.