Let's go through each part of the problem step-by-step.

(a) Show that $f(x) > 0$ for all real values of $x$.

The function given is:

$f(x) = x^2 - 8x + 18$

To prove that $f(x) > 0$ for all real values of $x$, we'll complete the square to find the minimum value of the quadratic function.

1. Complete the square:

   $f(x) = x^2 - 8x + 18$

   Rewrite the quadratic expression by isolating the constant term:

   $f(x) = (x^2 - 8x) + 18$

   To complete the square, take half of the coefficient of $x$, square it, and add/subtract it inside the parentheses:

   $\left( \frac{-8}{2} \right)^2 = (-4)^2 = 16$

   Add and subtract 16 inside the parentheses:

   $f(x) = (x^2 - 8x + 16) + 18 - 16$

   $f(x) = (x - 4)^2 + 2$

Now we have $f(x) = (x - 4)^2 + 2$. This expression shows that $f(x) \geq 2$ because $(x - 4)^2 \geq 0$ for all $x$.

Thus, the minimum value of $f(x)$ is 2, and $f(x) > 0$ for all real $x$.

(b) Express $f(x)$ in the form $f(x) = (x + p)^2 + q$.

From the completed square form obtained in part (a):

$f(x) = (x - 4)^2 + 2$

This is already in the form $f(x) = (x + p)^2 + q$, where:

$p = -4 \quad \text{and} \quad q = 2$

(c) Write down the equation of the parabola's axis of symmetry and the coordinates of its vertex.

From the form $f(x) = (x - 4)^2 + 2$:

1. Axis of symmetry:

   The axis of symmetry for a parabola of the form $f(x) = (x - h)^2 + k$ is given by the equation $x = h$. In this case, $h = 4$.

   So, the axis of symmetry is:

   $x = 4$

2. Vertex:

   The vertex of a parabola in the form $f(x) = (x - h)^2 + k$ is $(h, k)$. Here, $h = 4$ and $k = 2$.

   So, the coordinates of the vertex are:

   $(4, 2)$

Would you like more details or have any other questions?

Related Questions:

1. How do you find the axis of symmetry for a general quadratic equation?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How can you use the quadratic formula to find the roots of $f(x) = x^2 - 8x + 18$?
4. How would you determine the minimum or maximum value of a quadratic function without completing the square?
5. Can you derive the formula for the vertex of a quadratic equation using calculus?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same number inside the square.