Final answer:
a) The function f(x) = x^2 - 8x + 18 is always positive for all real values of x. b) The function can be expressed as f(x) = (x - 4)^2 + 2. c) The equation of the parabola's axis of symmetry is x = 4, and the coordinates of its vertex are (4, 2).
Step-by-step explanation:
(a) To prove that f(x) > 0 for all real values of x, we can use the fact that the function is a quadratic. Since the leading coefficient (the coefficient of x^2) is positive (1 in this case), the parabola opens upwards. This means that the vertex of the parabola is the lowest point on the graph, and since the vertex of this parabola is above the x-axis, it follows that the function is always positive.
(b) To express f(x) in the form f(x) = (x + p)^2 + q, we complete the square. We rewrite f(x) = x^2 - 8x + 18 as f(x) = (x^2 - 8x + 16) + 2. We can rewrite (x^2 - 8x + 16) as (x - 4)^2. Therefore, f(x) = (x - 4)^2 + 2.
(c) The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola. In this case, the equation of the axis of symmetry is x = 4. The x-coordinate of the vertex is the same as the value of p in the previous part, which is 4. Therefore, the coordinates of the vertex are (4, 2).