To identify the graph of the function \( f(x) = x^2 - 8x + 13 \), we can rewrite it in vertex form by completing the square.

1. Start with the original equation:
   \[
   f(x) = x^2 - 8x + 13
   \]

2. Complete the square:
   - Take half of the coefficient of \( x \) (which is \(-8\)), square it, and add/subtract it:
   \[
   f(x) = (x^2 - 8x + 16) - 16 + 13
   \]
   \[
   f(x) = (x - 4)^2 - 3
   \]

3. The vertex form of the quadratic is:
   \[
   f(x) = (x - 4)^2 - 3
   \]
   This shows that the vertex is at \( (4, -3) \).

### Key Features of the Graph:
- **Vertex:** \( (4, -3) \)
- **Direction:** Opens upwards (since the coefficient of \( x^2 \) is positive).
- **Y-intercept:** When \( x = 0 \):
  \[
  f(0) = 0^2 - 8(0) + 13 = 13
  \]
  So, the y-intercept is \( (0, 13) \).

### Summary of the Graph:
The graph is a parabola that opens upwards, with its vertex at \( (4, -3) \) and a y-intercept at \( (0, 13) \).

Would you like to see a plot of this function, or do you have any questions? Here are some related questions for further exploration:

1. How does the graph change if the coefficient of \( x^2 \) is negative?
2. What is the axis of symmetry for this function?
3. How do you find the x-intercepts of the graph?
4. Can you explain the significance of the vertex in the context of the graph?
5. What happens to the graph if we change the constant term from 13 to a larger number? 

**Tip:** Always convert quadratics into vertex form to easily identify key features like the vertex and direction of the graph.

Explore how to identify the graph of the quadratic function f(x) = x^2 - 8x + 13 by completing the square. Learn about its vertex, y-intercept, and key features. This content is designed for students in Grades 7-9.

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