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It looks like you've uploaded a set of functions. Let's break them down one by one and analyze the key characteristics like the roots, vertex (if applicable), and behavior of each function.

Here are the functions:

1. \( f(x) = x^2 - 12x + 32 \)
2. \( g(x) = x^2 + 2x - 3 \)
3. \( f(x) = x^2 - x \)
4. \( f(x) = x^2 + 5x - 2 \)

### 1. \( f(x) = x^2 - 12x + 32 \)
- **Factorization**: We will try to factor this or solve using the quadratic formula.
- **Roots**: Using either factorization or the quadratic formula will yield the roots (where the function crosses the x-axis).
- **Vertex**: The vertex form can be used to determine the maximum or minimum value of the parabola.

### 2. \( g(x) = x^2 + 2x - 3 \)
- **Factorization**: Look for two numbers that multiply to \(-3\) and add to \(2\).
- **Roots**: The roots give the x-intercepts.
- **Vertex**: Similar analysis to find the vertex.

### 3. \( f(x) = x^2 - x \)
- **Factorization**: We can factor this easily as \(x(x - 1)\).
- **Roots**: Setting \(x = 0\) and \(x - 1 = 0\) gives the roots.
- **Vertex**: Again, we can use the vertex formula.

### 4. \( f(x) = x^2 + 5x - 2 \)
- **Factorization**: Check for factorization or solve using the quadratic formula.
- **Roots**: Finding the points where \(f(x) = 0\).
- **Vertex**: Calculate the vertex using the formula for a parabola.

Would you like me to solve these functions in more detail and provide step-by-step solutions for roots, factorization, and vertices? Let me know how you'd like to proceed!

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Here are 5 related questions:
1. How do you find the vertex of a quadratic function?
2. What is the significance of the roots of a quadratic function?
3. Can all quadratics be factored easily? When do you use the quadratic formula?
4. How does the sign of the coefficient of \(x^2\) affect the direction of a parabola?
5. What role does completing the square play in converting a quadratic function?

**Tip:** The vertex of a parabola given by \(ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\).

Solve the following quadratic functions: f(x) = x^2 - 12x + 32, g(x) = x^2 + 2x - 3, f(x) = x^2 - x, f(x) = x^2 + 5x - 2

Learn how to solve quadratic functions like f(x) = x^2 - 12x + 32, g(x) = x^2 + 2x - 3 using methods such as factorization, finding roots, and vertex analysis. This tutorial provides step-by-step solutions suitable for Grades 9-11.

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