step 1 In decimal graph and calculator online, we can have the function f of x equals the absolute valu

Graph. f(x)=|x^2-1|

Name: Problem 81, Chapter 11: Elementary and Intermediate Algebra
Uploaded: 2020-05-25T13:23:58-08:00
Duration: 1 min 9 s
Channel: James Kiss
Description: Graph. f(x)=|x^2-1|

Key Concepts

Absolute Value Transformation

The absolute value transformation involves applying the absolute value function to another function. When a function is wrapped by an absolute value, all its output values become non-negative. This transformation can alter the original graph by reflecting portions where the function is negative, thus preserving the shape of the positive parts while flipping the negative parts upward.

Piecewise Definition

A function involving an absolute value can often be rewritten as a piecewise function, where different expressions define the function's behavior over different intervals. This is because the absolute value of an expression behaves differently depending on whether the inner expression is non-negative or negative, leading to distinct cases that must be considered to thoroughly understand and graph the function.

Quadratic Function

A quadratic function, typically represented in the form ax² + bx + c, has a parabolic shape and can have either a maximum or a minimum point known as the vertex. In the context of the given function, the quadratic function forms the basic underlying shape before the absolute value is applied, and understanding its properties, such as symmetry and the location of its vertex, is essential for a correct graphical representation.

Graphical Analysis of Functions

Graphical analysis involves understanding the structure of a function's graph by analyzing its algebraic form, transformations, and the effects of operations such as absolute value. This concept includes techniques for sketching the graph accurately by identifying key features like intercepts, turning points, and symmetry, and considering the impact of piecewise definitions on the overall graph.

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