Suppose f(x)=(1)/(2) x^2-8 for -4 ≤x ≤4, then the maximum value of the graph of |f(x)| is (A) -8

step 1 So here we have a function F of D, and this is going to be given to us by D squared plus 8D minu

Suppose f(x)=(1)/(2) x^2-8 for -4 ≤x ≤4, then the maximum...

Key Concepts

Quadratic Functions

Quadratic functions are second degree polynomials characterized by a parabolic shape and a formula of the form ax² + bx + c. Their graphs have a vertex, which represents either a maximum or a minimum point depending on the coefficient of x², and the axis of symmetry helps determine the behavior of the function across its domain.

Absolute Value Function

The absolute value function transforms any real number to its non-negative counterpart, reflecting negative outputs to positive ones while leaving positive values unchanged. When applied to another function, it affects the graph by turning all outputs non-negative, which can modify the locations of the maximum and minimum values compared to the original function.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function defined on a closed interval will have both a maximum and a minimum value. This principle is essential for analyzing the extremes of function outputs, especially when transformations like the absolute value are applied.

Domain Analysis

Domain analysis involves considering the specific interval over which a function is defined, which is crucial for determining the existence and location of extremal values. This analysis is important when a function is subject to boundary restrictions that influence its overall behavior, particularly in optimization problems.

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