The given function is \( f(x) = 2x^2 - 4x + 1 \).

### Key Features of the Quadratic Function:

1. **Standard Form:** The function is already in the standard quadratic form:
   \[
   f(x) = ax^2 + bx + c
   \]
   where \( a = 2 \), \( b = -4 \), and \( c = 1 \).

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

3. **Axis of Symmetry:** The axis of symmetry is the vertical line that passes through the vertex. Thus, it is:
   \[
   x = 1
   \]

4. **Direction of Opening:** Since the coefficient of \( x^2 \) is positive (i.e., \( a = 2 \)), the parabola opens **upward**.

5. **Y-intercept:** The y-intercept is the value of \( f(x) \) when \( x = 0 \):
   \[
   f(0) = 2(0)^2 - 4(0) + 1 = 1
   \]
   So, the y-intercept is \( (0, 1) \).

### Summary:
- Vertex: \( (1, -1) \)
- Axis of Symmetry: \( x = 1 \)
- Opens upward (parabola)
- Y-intercept: \( (0, 1) \)

Would you like further details or have any questions about this function?

Here are 5 questions to explore further:

1. How do we find the x-intercepts of this quadratic function?
2. What is the domain and range of \( f(x) = 2x^2 - 4x + 1 \)?
3. Can you graph the function using the key points found?
4. How would the function change if \( a \) were negative?
5. How do we complete the square to rewrite this function in vertex form?

**Tip:** The vertex form of a quadratic function can make it easier to identify the vertex and understand transformations.

Explore the key features of the quadratic function f(x) = 2x^2 - 4x + 1. Learn how to find the vertex, axis of symmetry, y-intercept, and the direction of the parabola. Perfect for high school students studying quadratic functions and parabolas.

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