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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.

Consider the function f(x) = x^2 - 8x + 18. (a) Show that f(x) > 0 for all real values of x. (b) Express f(x) in the form f(x) = (x + p)^2 + q. (c) The graph of f(x) is a parabola. Write down the equation of the parabola's axis of symmetry and the coordinates of its vertex.

Explore how to solve the quadratic function f(x) = x^2 - 8x + 18 using the method of completing the square. Learn how to express the function in the form (x + p)^2 + q, find the axis of symmetry, and determine the vertex of the parabola. Suitable for students in Grades 9-11.

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