Question

Which is the domain and range of the parabola with the equation y = 0.5(x2 – 12x – 6)?

Answer 1

The domain of the parabola is (-∞, ∞) and the range is (-∞, -222].

Step-by-step explanation:

The equation of the parabola is y = 0.5(x² – 12x – 6). To find the domain and range, we need to analyze the behavior of the parabola.

The domain is the set of all possible x-values for the parabola. Since a parabola is defined for all real numbers, the domain is (-∞, ∞).

The range is the set of all possible y-values for the parabola. To find the range, we can analyze the vertex and the direction of the parabola. The x-coordinate of the vertex is given by x = -b/2a, which in this case is x = -(-12)/(2*0.5) = 12. The y-coordinate can be found by substituting this x-value into the equation: y = 0.5(12² – 12*12 – 6) = -222. Therefore, the range is (-∞, -222].

Answer 2

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].