Question

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)=−x2+14x−106f(x) = $$-x^2 + 14x - 106$$

Accepted Answer

Identify the type of function given. Since the function is $$f(x) = -x^2$ + 14x - 106$$, it is a quadratic function with a negative leading coefficient, which means its graph is a parabola opening downward and it has a maximum value. Find the vertex of the parabola, since the vertex gives the maximum or minimum value of a quadratic function. Use the vertex formula for the x-coordinate: $$x = \frac{-b}{2a}$$, where $$a = -1$$ and $$b = 14$$ from the function $$f(x) = ax^2$ + bx + c. $$Calculate the y-coordinate of the vertex by substituting the x-value found into the original function: $$f(x) = -x^2$ + 14x - 106. $$This y-value is the maximum value of the function. Determine the domain of the function. Since it is a quadratic function, the domain is all real numbers, which can be written as $$(-\infty, \infty). $$Determine the range of the function. Because the parabola opens downward and the vertex represents the maximum value, the range is all real numbers less than or equal to the maximum y-value found at the vertex. Express the range as $$(-\infty, 	ext{maximum value}].$$

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. $f (x) = - x^{2} + 14 x - 106$

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Concetti chiave

Quadratic Functions and Their Graphs

Vertex of a Parabola

Domain and Range of Quadratic Functions

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)=−x2+14x−106f(x) = -x^2 + 14x - 106