True or false: The function defined by g(x)=-5 x^2 has a...

Name: Problem 79, Chapter 7: Intermediate Algebra
Uploaded: 2020-06-23T06:06:49-08:00
Duration: 30 s
Channel: Ankit Gupta
Description: True or false: The function defined by g(x)=-5 x^2 has a maximum value but no minimum value.

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Key Concepts

Concavity and Extreme Values

The concavity of a quadratic function, determined by the sign of the leading coefficient, indicates whether the vertex represents a maximum or a minimum. If the coefficient is positive, the parabola opens upwards and the vertex is a minimum; if the coefficient is negative, as in this case, the parabola opens downwards and the vertex is a maximum. This property explains why some quadratic functions have a highest value but no lowest bound.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form ax² + bx + c. They form the basis for many algebraic and geometric concepts because their graphs, which are parabolas, provide clear examples of how algebraic equations can represent curved shapes in the plane.

Vertex of a Parabola

The vertex of a quadratic function's graph is the point where the function achieves its extreme value—either a maximum or a minimum. This point, determined by the coefficients of the quadratic equation, is crucial for understanding the function's behavior and for solving optimization problems.

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