Final answer:
The domain of the parabola is (-∞, ∞) and the range is (-∞, -78].
Step-by-step explanation:
The equation of the parabola is given by y = 0.5(x^2 – 12x – 6).
To determine the domain of the parabola, we need to find the x-values for which the equation is defined. Since this is a quadratic equation, it is defined for all real numbers. Therefore, the domain is (-∞, ∞).
The range of the parabola can be determined by analyzing the coefficient of the x^2 term. In this case, the coefficient is positive, which means the parabola opens upwards. Hence, the minimum value of the y-coordinate occurs at the vertex of the parabola. To find the vertex, we can use the formula x = -b/(2a), where a and b are the coefficients of the x^2 and x terms, respectively. Plugging in the values, we get x = -(-12) / (2 * 0.5) = 12.
Substituting the value of x into the equation, we get y = 0.5(12^2 – 12(12) – 6) = -78.
Therefore, the range of the parabola is (-∞, -78].