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 94

Capitolo 3, Problema 94

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

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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) and symmetry about the y-axis. The graph increases linearly for x > 0 and decreases linearly for x < 0.

Identify the transformations applied to f(x) = |x| to obtain g(x) = -2|x+3|+2. The transformations include: (1) a horizontal shift, (2) a vertical stretch and reflection, and (3) a vertical shift.

Apply the horizontal shift: The term |x+3| indicates a shift 3 units to the left. This moves the vertex of the graph from (0, 0) to (-3, 0).

Apply the vertical stretch and reflection: The coefficient -2 in front of |x+3| stretches the graph vertically by a factor of 2 and reflects it across the x-axis. This makes the V-shape open downward and steeper.

Apply the vertical shift: The +2 at the end of the function shifts the entire graph 2 units upward. The new vertex of the graph is at (-3, 2). Combine all these transformations to sketch the final graph of g(x).

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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) = -2|x+3|+2, the transformations include a horizontal shift left by 3 units, a vertical stretch by a factor of 2, and a reflection across the x-axis.

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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 final graph and understand the relationship between the original and transformed functions.

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Graphs and Coordinates - Example

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