To sketch the graph of the function \( y = x^2 - 4x + 4 \), let's analyze its key features, such as the vertex, axis of symmetry, intercepts, and overall shape. 

### Step-by-Step Analysis

1. **Identify the Type of Function**:  
   The function \( y = x^2 - 4x + 4 \) is a quadratic function. Its general form is \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -4 \), and \( c = 4 \).

2. **Determine the Shape**:  
   Since \( a = 1 > 0 \), the parabola opens upwards.

3. **Find the Vertex**:  
   The vertex form of a parabola is given by \( x = -\frac{b}{2a} \). For this function:
   \[
   x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2.
   \]
   To find the y-coordinate of the vertex, substitute \( x = 2 \) into the function:
   \[
   y = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0.
   \]
   So, the vertex is at \( (2, 0) \).

4. **Axis of Symmetry**:  
   The axis of symmetry for a parabola \( y = ax^2 + bx + c \) is the vertical line passing through the vertex. In this case, it is \( x = 2 \).

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

6. **Find the X-intercepts**:  
   The x-intercepts occur when \( y = 0 \). We solve the quadratic equation:
   \[
   x^2 - 4x + 4 = 0.
   \]
   Factor the quadratic:
   \[
   (x - 2)(x - 2) = 0.
   \]
   So, \( x = 2 \). This confirms that there is a **double root** at \( x = 2 \). The parabola touches the x-axis at \( (2, 0) \) but does not cross it.

7. **Sketch the Graph**:  
   - Plot the vertex at \( (2, 0) \).
   - The parabola opens upwards.
   - The axis of symmetry is the vertical line \( x = 2 \).
   - Plot the y-intercept at \( (0, 4) \).
   - Since there is a double root at \( x = 2 \), the graph just touches the x-axis at this point.

### Graph of the Function
Based on the analysis above, the graph is a parabola that touches the x-axis at \( (2, 0) \) and has a vertex at the same point. It opens upwards and has a y-intercept at \( (0, 4) \).

If you'd like a visual representation, here is a sketch of the graph:

```
  y
  |
  |               *
  |             *   
  |          *     
  |       *   
  |     *   
  |  *        
--+------------------ x
  |     *        
  |       *        
  |          *        
  |             *   
  |               *
  |
```

Would you like more details on any step, or have any specific questions?

### Related Questions:

1. How would the graph change if the equation were \( y = x^2 - 4x + 5 \)?
2. What are the properties of a quadratic function?
3. How do you find the vertex of a quadratic function in standard form?
4. What is the significance of the axis of symmetry in a parabola?
5. How does the value of \( a \) affect the direction and width of a parabola?

**Tip:** The vertex form of a quadratic function, \( y = a(x-h)^2 + k \), makes it easy to identify the vertex and direction of the parabola.

Learn how to sketch the graph of the quadratic function Y= x^(2) - 4x + 4 with detailed step-by-step analysis. This problem covers key concepts in quadratic functions, including vertex determination, axis of symmetry, intercepts, and the shape of the graph. Suitable for students in Grades 7-9.

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