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Chapter 3: Problem 42

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)=-2 x^{2}-12 x+3$$

Short Answer

Expert verified
The given quadratic function \(f(x) = -2x^{2}-12 x+3\) has a maximum value of -27 which is reached at x = 3. The function's domain is all real numbers, and the range is all y such that \(y 鈮� -27\).

Step by step solution

01

Determine the orientation of the parabola and whether the function has a maximum or minimum value

A quadratic function opens upward if the coefficient of \(x^{2}\) (a) is positive and opens downward if a is negative. Here, a = -2 which is less than zero. Hence the parabola opens downward, indicating that the function has a maximum value.
02

Compute coordinates of the vertex to find maximum value

The maximum or minimum value of the quadratic function occurs at its vertex, which is given by the formula \(h = -\frac{b}{2a}\). For the function \(f(x)=-2 x^{2}-12 x+3\), b=-12 and a=-2. Using these values in the formula, the x-coordinate of the vertex (h) can be computed as \(h = -\frac{-12}{2*-2}= 3\). The maximum value of the function is given by the y-coordinate of the vertex, which is obtained by substituting this 'h' back into the function to compute f(h), gives \(f(3)= -2*3^{2}-12*3+3=-27\). Hence, the maximum value is -27 and it occurs at x=3.
03

Identify the function's domain and range

For any quadratic function, the domain is all real numbers since x (the input) can be any real number. The range depends upon whether the function has a maximum or minimum value. In this case, as we previously established that the function has a maximum value, the range is \(y 鈮� -27\).

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