The function given is \( f(x) = x^2 - 4x + 1 \). Let's analyze it step by step.

### 1. **Vertex of the Parabola:**
The function is quadratic, and its graph is a parabola. The vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For the equation \( f(x) = ax^2 + bx + c \), we have:
- \( a = 1 \)
- \( b = -4 \)
- \( c = 1 \)

So,
\[
x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2
\]

Substitute \( x = 2 \) back into the function to find \( y \):
\[
f(2) = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3
\]

So, the vertex is \( (2, -3) \).

### 2. **Axis of Symmetry:**
The axis of symmetry is the vertical line that passes through the vertex:
\[
x = 2
\]

### 3. **Y-Intercept:**
The y-intercept is the point where \( x = 0 \):
\[
f(0) = (0)^2 - 4(0) + 1 = 1
\]
So, the y-intercept is \( (0, 1) \).

### 4. **X-Intercepts:**
To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[
x^2 - 4x + 1 = 0
\]
Use the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}
\]

So, the x-intercepts are \( x = 2 + \sqrt{3} \) and \( x = 2 - \sqrt{3} \).

### 5. **Graphing Information:**
The parabola opens upwards because the coefficient of \( x^2 \) is positive (1). The vertex \( (2, -3) \) is the lowest point on the graph, and the function decreases until \( x = 2 \) and increases thereafter.

---

Would you like more details on any part of this analysis? Here are some related questions:

1. What is the general process for finding the vertex of a quadratic function?
2. How do you derive the axis of symmetry from a quadratic equation?
3. What role does the discriminant play in determining the nature of the roots?
4. How does the sign of the leading coefficient affect the graph of a parabola?
5. Can the quadratic formula be used for all quadratic equations?
6. How do you complete the square to convert the equation into vertex form?
7. What are the applications of quadratic functions in real-world problems?
8. How do you determine if a parabola has real or complex roots?

**Tip:** Always check the discriminant (\( b^2 - 4ac \)) when solving quadratic equations. It reveals if the roots are real or complex.

Explore the analysis of the quadratic function f(x) = x² - 4x + 1, including finding the vertex, axis of symmetry, y-intercept, x-intercepts, and graphing information. Understand key algebraic concepts and formulas involved. Suitable for Grades 9-12.

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