Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. \[ f(x)=x^{2}-8 x+12 \] (a) The vertex is (Type an ordered pair, using integers or fractions.) (b) Determine whether the parabola has a maximum value or a minimum value and find the value. Select the correct choice below and fill in the answer box within your choice. (Type an integer or a fraction.) A. The parabola opens upward and has a minimum value of B. The parabola opens downward and has a maximum value of (c) What is the range of $f(x)$ ? The range of $f(x)$ is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) (d) On what interval is the function increasing? The function is increasing on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) On what interval is the function decreasing? The function is decreasing on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Step 1 ：The given function is a quadratic function in the form of \(f(x) = ax^2 + bx + c\), where \(a = 1\), \(b = -8\), and \(c = 12\).

Step 2 ：The vertex of a parabola \(y = ax^2 + bx + c\) is given by the point \((-b/2a, f(-b/2a))\).

Step 3 ：Since the coefficient of \(x^2\) is positive, the parabola opens upward and has a minimum value at the vertex.

Step 4 ：The range of the function is all the y-values that the function can take. Since the parabola opens upward and has a minimum value, the range is \([f(-b/2a), +\infty)\).

Step 5 ：The function is increasing on the interval \((-b/2a, +\infty)\) and decreasing on the interval \((-\infty, -b/2a)\).

Step 6 ：Calculating these values, we find that the vertex is \((4, -4)\).

Step 7 ：The parabola opens upward and has a minimum value of \(-4\).

Step 8 ：The range of \(f(x)\) is \([-4, +\infty)\).

Step 9 ：The function is increasing on the interval \((4, +\infty)\) and decreasing on the interval \((-\infty, 4)\).

Step 10 ：\(\boxed{\text{Final Answer:}}\)

Step 11 ：\(\boxed{\text{(a) The vertex is (4, -4).}}\)

Step 12 ：\(\boxed{\text{(b) The parabola opens upward and has a minimum value of -4.}}\)

Step 13 ：\(\boxed{\text{(c) The range of } f(x) \text{ is } [-4, +\infty).}\)

Step 14 ：\(\boxed{\text{(d) The function is increasing on the interval } (4, +\infty) \text{ and decreasing on the interval } (-\infty, 4).}\)