Answer: To find the range of the quadratic function y = x^2 - 8x + 12, we can use different methods, but one way is to complete the square and express the function in vertex form.
y = x^2 - 8x + 12 (given function)
y = (x - 4)^2 - 4 (completing the square)
The vertex of the parabola represented by this function is at the point (4,-4). This means that the minimum value of the function occurs when x=4 and y=-4. Since the parabola opens upward, this minimum value is also the lowest point of the parabola and is the lower bound of the range.
Thus, the range of the function is all real numbers greater than or equal to -4, i.e., the range is (-4, infinity).
Another way to see this is to note that the coefficient of x^2 in the given function is positive, which means that the parabola opens upward and has a minimum value. Therefore, the range of the function is all real numbers greater than or equal to the y-coordinate of the vertex, which is -4.
Explanation: