In Exercises 5–6, 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 + 14x - 106

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1. Identify the type of function: The given function \( f(x) = -x^2 + 14x - 106 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = -1 \), \( b = 14 \), and \( c = -106 \).
2. Determine if the function has a maximum or minimum value: Since the coefficient \( a = -1 \) is negative, the parabola opens downwards, indicating that the function has a maximum value.
3. Find the vertex of the parabola: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). Substitute \( b = 14 \) and \( a = -1 \) into the formula to find the x-coordinate of the vertex.
4. Calculate the maximum value: Substitute the x-coordinate of the vertex back into the function \( f(x) \) to find the maximum value of the function.
5. Determine the domain and range: The domain of any quadratic function is all real numbers, \( (-\infty, \infty) \). The range is determined by the maximum value, so it will be \( (-\infty, \text{maximum value}] \).

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the properties of quadratic functions is essential for determining their maximum or minimum values.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point of the graph, depending on its orientation. For a quadratic function in standard form, the vertex can be found using the formula x = -b/(2a). This point is crucial for identifying the maximum or minimum value of the function, as it occurs at the vertex when the parabola opens downwards or upwards, respectively.

Domain and Range

The domain of a function refers to all possible input values (x-values) that the function can accept, while the range refers to all possible output values (f(x)-values) that the function can produce. For quadratic functions, the domain is typically all real numbers, but the range is determined by the vertex and the direction the parabola opens, which affects the minimum or maximum output value.