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The image contains a set of quadratic functions for which you need to determine the vertex and the points where they intersect the x-axis (the roots). Additionally, there is a summary of steps to find these values.

### List of Quadratic Functions:
1. \( f(x) = x^2 - 8x + 12 \)
2. \( f(x) = x^2 - 5x - 24 \)
3. \( f(x) = x^2 - x - 20 \)
4. \( f(x) = x^2 - 12x + 27 \)
5. \( f(x) = x^2 - x - 30 \)
6. \( f(x) = x^2 + 16x + 63 \)
7. \( f(x) = x^2 - 8x + 12 \)
8. \( f(x) = x^2 - 6x + 5 \)
9. \( f(x) = x^2 - 13x + 42 \)
10. \( f(x) = x^2 + x - 2 \)
11. \( f(x) = 3x^2 - 2x - 8 \)
12. \( f(x) = 4x^2 + 8x + 3 \)
13. \( f(x) = 2x^2 + 11x + 5 \)
14. \( f(x) = 20x^2 + 7x - 6 \)
15. \( f(x) = 3x^2 + 7x - 6 \)

### Summary of Steps to Solve:
1. **Find the vertex:**  
   The x-coordinate of the vertex can be found using the formula:  
   \[ x = -\frac{b}{2a} \]  
   where \( a \) and \( b \) are coefficients from the standard form of the quadratic equation \( ax^2 + bx + c \).

2. **Find the y-coordinate of the vertex:**  
   Substitute the x-coordinate back into the function \( f(x) \) to find the y-coordinate.

3. **Solve the function for roots (x-intercepts):**  
   Set the function to zero: \( f(x) = 0 \), and solve for \( x \) using methods such as factoring, completing the square, or using the quadratic formula:  
   \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

4. **Find the points where the function crosses the x-axis:**  
   The solutions for \( x \) from the quadratic formula will give you the x-coordinates of the points \( P_1(x_1, 0) \) and \( P_2(x_2, 0) \).

5. **Graph the function:**  
   Plot the vertex and the points where the function crosses the x-axis to sketch the graph.

Would you like me to solve one of these functions step-by-step, or provide details on all of them? If so, which one should I start with?

### Questions for Further Exploration:
1. How do the roots of a quadratic equation relate to its graph?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How does changing the coefficients \(a\), \(b\), and \(c\) affect the graph of a quadratic function?
4. What are some real-life applications of quadratic functions?
5. What methods can be used to solve a quadratic equation if it can't be factored easily?

**Tip:** Always check the discriminant \(b^2 - 4ac\) to determine the nature of the roots before solving a quadratic equation.

Determine the vertex and points of intersection of the following quadratic functions.

Learn how to determine the vertex and the x-intercepts of various quadratic functions, such as x^2 - 8x + 12 and others. This guide covers key algebraic concepts and formulas, ideal for students in Grades 9-11.

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