To determine the domain and range of the parabola given by the equation [tex]\( y = 0.5(x^2 - 12x - 6) \)[/tex], we need to follow a series of steps.
1. Determine the Domain:
The domain of a quadratic function (parabola) is all real numbers because the quadratic function is defined for all [tex]\( x \)[/tex] values. Therefore, the domain is [tex]\( (-\infty, \infty) \)[/tex].
2. Determine the Range:
To find the range, we need to identify the vertex of the parabola, as it gives us the minimum or maximum value depending on the orientation of the parabola.
The standard form of a parabolic equation is [tex]\( y = ax^2 + bx + c \)[/tex].
For our equation [tex]\( y = 0.5(x^2 - 12x - 6) \)[/tex], we identify:
[tex]\[
a = 0.5, \quad b = -12, \quad c = -6
\][/tex]
The x-coordinate of the vertex [tex]\( h \)[/tex] can be found using the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
h = -\frac{-12}{2 \times 0.5} = \frac{12}{1} = 12
\][/tex]
To find the y-coordinate [tex]\( k \)[/tex] of the vertex, substitute [tex]\( x = 12 \)[/tex] into the original equation:
[tex]\[
y = 0.5(12^2 - 12 \cdot 12 - 6) \][/tex]
First, simplify inside the parentheses:
[tex]\[
12^2 - 12 \cdot 12 - 6 = 144 - 144 - 6 = -6
\][/tex]
Then, multiply by 0.5:
[tex]\[
y = 0.5 \cdot (-6) = -78
\][/tex]
So the vertex is [tex]\( (12, -78) \)[/tex].
Since [tex]\( a > 0 \)[/tex] (0.5 in this case), the parabola opens upwards.
Thus, the range of the parabola is from the y-coordinate of the vertex to infinity:
[tex]\[
\text{Range} = [-78, \infty)
\][/tex]
Summarizing, we have:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( [-78, \infty) \)[/tex]
From the provided options, the correct answer is:
[tex]\[ \boxed{D : (-\infty, \infty) \quad \text{;}\quad R:[-78, \infty)} \][/tex]
