https://file.aiquickdraw.com/mathbot/file/1727931790910859fdo3i.jpg

Let's break down the questions visible in the image:

1. **Part a** asks to graph the function \( f(x) = x^2 + 5x - 6 \) by hand.
   - This is a quadratic equation of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 5 \), and \( c = -6 \). The graph will be a parabola.
   - To graph the function:
     - Find the vertex using \( x = -\frac{b}{2a} = -\frac{5}{2(1)} = -2.5 \).
     - Calculate the value of \( f(x) \) at \( x = -2.5 \).
     - Identify the intercepts (roots) by factoring or using the quadratic formula.

2. **Part b** asks to solve the inequality \( x^2 - 6x + 9 \geq 0 \).
   - This is a perfect square: \( (x - 3)^2 \geq 0 \).
   - Since the square of any real number is non-negative, \( (x - 3)^2 \geq 0 \) for all \( x \), with equality when \( x = 3 \).
   - Therefore, the solution set is \( x \in (-\infty, \infty) \), or more specifically, \( [3, \infty) \) when focusing on the critical point.

For the **graph choice** problem:
- The function's graph appears to be related to the inequality or the quadratic function from the earlier part. Based on the function \( f(x) = x^2 + 5x - 6 \), the correct graph should be a parabola opening upwards with roots at \( x = -6 \) and \( x = 1 \).
  
Based on this, the correct graph seems to be **Option B** (but I would need to verify each graph's details to confirm).

If you'd like further explanations, clarifications, or specific steps, feel free to ask! Also, let me know if you need assistance with part b of the graph problem.

### Related Questions:
1. How do you find the x-intercepts of a quadratic function?
2. What is the significance of the vertex in a parabola?
3. How do you solve a quadratic inequality graphically?
4. Can quadratic equations always be factored?
5. How does completing the square help in solving quadratic equations?

### Tip:
Always check whether a quadratic expression is factorable before using the quadratic formula—it can save time!

Graph the function f(x) = x^2 + 5x - 6 and solve f(x) >= 0.

Learn how to graph the quadratic equation x^2 + 5x - 6 and solve the inequality f(x) >= 0 step-by-step. This problem focuses on key algebraic concepts such as graphing parabolas, finding intercepts, and solving quadratic inequalities. Ideal for students in Grades 9-11.

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