- The given equation is a quadratic function \(y = 0.5(x^2 - 8x - 6)\), which is a polynomial defined for all real numbers.
- The domain of this quadratic function is all real numbers, denoted as \((-\infty, \infty)\).
- Using the vertex formula \(x = -b / (2a)\), we calculate the vertex x-coordinate to be \(4\) and the corresponding y-coordinate to be \(-11\).
- Since the parabola opens upward, the range is \([-11, \infty)\).
- Final answer: \(\boxed{D:(-\infty, \infty); R:[-11, \infty)}\).
Explanation
- Analysis of the Function and Domain
The given function is a quadratic equation of the form \(y = 0.5(x^2 - 8x - 6)\). By distributing the \(0.5\), we can rewrite this as \(y = 0.5x^2 - 4x - 3\). This is a polynomial function of degree 2, commonly known as a parabola. Any polynomial function is defined for all real numbers, so the domain is \((-\infty, \infty)\).
- Finding the Vertex of the Parabola
To find the range, we identify the vertex of the parabola. For a quadratic function \(y = ax^2 + bx + c\), the x-coordinate of the vertex is given by the formula \(x = -b / (2a)\). Here, \(a = 0.5\) and \(b = -4\), so the x-coordinate of the vertex is \(x = -(-4) / (2 \times 0.5) = 4 / 1 = 4\). To find the y-coordinate of the vertex, substitute \(x = 4\) into the original equation: \(y = 0.5(4^2 - 8(4) - 6) = 0.5(16 - 32 - 6) = 0.5(-22) = -11\). Thus, the vertex of the parabola is at \((4, -11)\).
- Determining the Range and Concluding
Since the coefficient of \(x^2\) is \(a = 0.5\), which is positive, the parabola opens upwards. This means that the vertex represents the minimum point of the function. Therefore, the function takes all values greater than or equal to the y-coordinate of the vertex. The range of the function is \([-11, \infty)\). Combining our findings, the domain is \((-\infty, \infty)\) and the range is \([-11, \infty)\). Comparing this to the given options, the correct choice is the second one.
Examples
Quadratic functions are used to model various real-world scenarios where there is a maximum or minimum value. For example, if you are throwing a ball, the height of the ball over time can be modeled by a quadratic equation. The vertex of the parabola represents the peak height of the ball, which helps in understanding the maximum height achieved during its flight. In our case, the vertex at \(-11\) indicates the lowest point of a trajectory or a cost function. Understanding the domain and range is essential in engineering and physics to ensure that models remain within realistic bounds.