The given equation is:

\[
y = x^2 - 4x + 1
\]

This is a quadratic equation in standard form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -4 \), and \( c = 1 \).

### 1. Vertex Form and Vertex of the Parabola
To find the vertex of the parabola, we can complete the square or use the vertex formula:

\[
x_{\text{vertex}} = \frac{-b}{2a} = \frac{-(-4)}{2(1)} = 2
\]

Now, substitute \( x = 2 \) into the original equation to find the \( y \)-coordinate of the vertex:

\[
y_{\text{vertex}} = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3
\]

Thus, the vertex is at \( (2, -3) \).

### 2. Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex. Therefore, the equation of the axis of symmetry is:

\[
x = 2
\]

### 3. Direction of the Parabola
Since the coefficient of \( x^2 \) is positive (\( a = 1 \)), the parabola opens **upward**.

### 4. Y-Intercept
To find the y-intercept, set \( x = 0 \):

\[
y = (0)^2 - 4(0) + 1 = 1
\]

Thus, the y-intercept is \( (0, 1) \).

### 5. X-Intercepts
To find the x-intercepts, set \( y = 0 \) and solve for \( x \):

\[
0 = x^2 - 4x + 1
\]

We can solve this using 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:

\[
x = 2 + \sqrt{3}, \quad x = 2 - \sqrt{3}
\]

---

Would you like more details on any of the steps or concepts? Here are some related questions:

1. How do you complete the square for the given quadratic equation?
2. What is the general method for solving a quadratic equation?
3. Can you graph the parabola for the given equation?
4. How does the discriminant help determine the nature of the roots of a quadratic equation?
5. How can you rewrite the equation in vertex form?

**Tip:** The vertex form of a quadratic equation \( y = a(x-h)^2 + k \) provides an easy way to determine the vertex and direction of the parabola.

This page provides a comprehensive guide to solving the quadratic equation y = x^2 - 4x + 1. Explore how to find the vertex, axis of symmetry, y-intercept, and x-intercepts using the quadratic formula and vertex theorem. Ideal for students in Grades 9-11 looking to master quadratic equations and graphing parabolas.

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