The function you've provided is:

\[
f(x) = x^2 - 4x + 5
\]

This is a quadratic function. Let's break it down:

1. **Form**: It follows the general quadratic form \( f(x) = ax^2 + bx + c \), where:
   - \( a = 1 \)
   - \( b = -4 \)
   - \( c = 5 \)

2. **Vertex**: The vertex of a quadratic function can be found using the formula \( x = \frac{-b}{2a} \). Plugging in \( a = 1 \) and \( b = -4 \):
   \[
   x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2
   \]
   To find the y-coordinate of the vertex, substitute \( x = 2 \) into the original function:
   \[
   f(2) = (2)^2 - 4(2) + 5 = 4 - 8 + 5 = 1
   \]
   So, the vertex is \( (2, 1) \).

3. **Axis of Symmetry**: The axis of symmetry is the vertical line through the vertex, so it is \( x = 2 \).

4. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
   \[
   f(0) = 0^2 - 4(0) + 5 = 5
   \]
   So the y-intercept is \( (0, 5) \).

5. **Minimum value**: Since the parabola opens upwards (because \( a = 1 > 0 \)), the vertex represents the minimum point of the function. The minimum value of \( f(x) \) is \( f(2) = 1 \).

Would you like more details or further clarification?

Here are 5 related questions to expand on this:
1. How do you find the roots of \( f(x) = x^2 - 4x + 5 \)?
2. What is the domain and range of this function?
3. How do you graph a quadratic function step by step?
4. How can you complete the square to rewrite \( f(x) = x^2 - 4x + 5 \)?
5. How would the function change if \( a \) were negative?

**Tip**: The vertex form of a quadratic equation makes it easier to graph the function and understand its transformations.

Explore the quadratic function f(x) = x^2 - 4x + 5 with a step-by-step breakdown of the vertex, axis of symmetry, y-intercept, and minimum value. This problem covers key algebraic concepts and is suitable for students in Grades 8-10.

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