"Graph the equation y = x^2 - 4x + 4 and find its vertex and intercepts using the eMath graphing solver."

I can't directly access or utilize external tools like the eMath graphing solver, but I can help you graph the equation, find the vertex, and determine the intercepts.

### 1. **Equation:**
   \[
   y = x^2 - 4x + 4
   \]

### 2. **Finding the Vertex:**
   The vertex of a quadratic equation in the form \( y = ax^2 + bx + c \) can be found using the formula:
   \[
   x = \frac{-b}{2a}
   \]
   Here, \( a = 1 \), \( b = -4 \), and \( c = 4 \).

   \[
   x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2
   \]

   Plug this value of \( x \) back into the equation to find \( y \):
   \[
   y = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0
   \]

   So, the vertex is at \( (2, 0) \).

### 3. **Finding the y-intercept:**
   The y-intercept occurs where \( x = 0 \):
   \[
   y = (0)^2 - 4(0) + 4 = 4
   \]
   So, the y-intercept is at \( (0, 4) \).

### 4. **Finding the x-intercepts:**
   The x-intercepts occur where \( y = 0 \). Set the equation equal to zero:
   \[
   x^2 - 4x + 4 = 0
   \]
   This can be factored as:
   \[
   (x - 2)^2 = 0
   \]
   So, \( x = 2 \).

   Thus, the x-intercept is at \( (2, 0) \), which is also the vertex.

### 5. **Graphing the Equation:**
   The equation \( y = x^2 - 4x + 4 \) is a parabola that opens upwards. It touches the x-axis at \( (2, 0) \) and crosses the y-axis at \( (0, 4) \).

Would you like more details or have any questions? Here are some related problems you might find interesting:

1. How do you find the vertex of a parabola using the vertex form of a quadratic equation?
2. What is the significance of the discriminant in determining the nature of the roots of a quadratic equation?
3. How can you convert a quadratic equation from standard form to vertex form?
4. How does the graph of \( y = ax^2 + bx + c \) change as the values of \( a \), \( b \), and \( c \) change?
5. What is the axis of symmetry for a parabola, and how is it related to the vertex?
6. How do you determine the direction in which a parabola opens?
7. What are the steps to find the focus and directrix of a parabola?
8. How does completing the square help in finding the vertex form of a quadratic equation?

**Tip:** When graphing a quadratic equation, identifying the vertex and axis of symmetry can greatly simplify the process.

Learn how to graph the equation y = x^2 - 4x + 4, find its vertex at (2, 0), y-intercept at (0, 4), and x-intercept at (2, 0). Understand the key formulas and concepts involved in solving quadratic equations. Suitable for high school students.

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