In Exercises 17–18, use the function’s equation, and not its...

In Exercises 17–18, use the function’s equation, and not its graph, to find
a. the minimum or maximum value and where it occurs.
b. the function’s domain and its range.
$$
f(x)=-x^{2}+14 x-106
$$

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically expressed in the form ax² + bx + c. Their graphs are parabolas, which open upward if the coefficient a is positive and downward if a is negative. This determines whether the function has a minimum or a maximum value.

Vertex of a Parabola

The vertex is the point on the graph of a quadratic function where the maximum or minimum value occurs. It can be found using the formula x = -b/(2a) for the x-coordinate, and substituting this back into the function to get the corresponding y-coordinate. The vertex is critical for understanding the function's extreme values.

Domain and Range

The domain of a quadratic function is all real numbers because every real input has a corresponding output. The range is the set of possible output values, which is limited by the vertex. For a parabola opening upward, the range starts at the minimum y-coordinate of the vertex and extends to infinity, whereas for a parabola opening downward, it extends from negative infinity to the maximum y-coordinate.

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