To sketch the graph of the quadratic function $y = x^2 - 4x + 4$, let's break down the key features of the function.

1. Identify the type of function:

• The equation $y = x^2 - 4x + 4$ is a quadratic function, which means its graph will be a parabola.

2. Rewrite the function in vertex form:

• The standard form is $y = ax^2 + bx + c$. Here, $a = 1$, $b = -4$, and $c = 4$.
• The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Complete the square to rewrite the function in vertex form: $y = x^2 - 4x + 4 = (x - 2)^2$
• The vertex is $(2, 0)$.

3. Determine the axis of symmetry:

• The axis of symmetry is a vertical line that passes through the vertex. For this function, it's $x = 2$.

4. Identify the direction of the parabola:

• Since $a = 1 > 0$, the parabola opens upwards.

5. Find the y-intercept:

• The y-intercept occurs where $x = 0$. $y = 0^2 - 4(0) + 4 = 4$
• The y-intercept is $(0, 4)$.

6. Find the x-intercepts:

• The x-intercepts occur where $y = 0$. $0 = (x - 2)^2$
• Solve for $x$: $x = 2$.
• The x-intercept is a double root at $(2, 0)$, meaning the parabola touches the x-axis at this point.

7. Sketch the graph:

• Draw a parabola with the vertex at $(2, 0)$ and passing through the point $(0, 4)$.
• The parabola touches the x-axis at $(2, 0)$ and opens upwards.

Here is a simple sketch of the graph:

y
|
|       *
|     *   *
|    *     *
|   *       *
|  *         *
| *           *
------------------------- x
            (2, 0)

Do you want details or have any questions? Here are 5 related questions to further explore:

1. How do you determine if a quadratic function has a maximum or minimum point?
2. How would the graph change if the coefficient of $x^2$ was negative?
3. What is the importance of the discriminant in determining the nature of the roots?
4. How do you find the vertex of a quadratic function using the formula $-\frac{b}{2a}$?
5. How can you determine the number of x-intercepts a quadratic function will have?

Tip: When sketching graphs of quadratic functions, always start by identifying the vertex, axis of symmetry, and the direction in which the parabola opens. This provides a solid foundation for the rest of the graph.