Question

determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = -2x² + 12x the quadratic function has a ▼ value. the value is □.

Explanation:

Step1: Determine max/min (a sign)

For \( f(x) = ax^2 + bx + c \), \( a = -2 \). Since \( a < 0 \), parabola opens down, so maximum value.

Step2: Find vertex x-coordinate

Vertex x-coordinate: \( x = -\frac{b}{2a} \). Here, \( b = 12 \), \( a = -2 \). So \( x = -\frac{12}{2(-2)} = \frac{-12}{-4} = 3 \).

Step3: Find max value (sub x=3)

Sub \( x = 3 \) into \( f(x) \): \( f(3) = -2(3)^2 + 12(3) = -2(9) + 36 = -18 + 36 = 18 \).

Answer:

The quadratic function has a maximum value. The value is 18.