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.