91Ó°ÊÓ

A quadratic function is a polynomial function with a degree of two, and it is usually expressed in the form \( f(x) = ax^2 + bx + c \). This simple yet powerful function forms a parabola when graphed on the Cartesian plane.
The parabola could open upwards if the coefficient \( a \) is positive, or downwards if \( a \) is negative. The highest or lowest point on this graph is called the vertex.

• The coefficient \( a \) determines the width and direction of the parabola.
• The coefficient \( b \) affects the position and orientation of the parabola.
• The constant \( c \) is the y-intercept, the point where the graph intersects the y-axis.

Understanding these parameters for any quadratic function helps us predict its behavior.
In our exercise, the function \( f(z) = z^2 - 2z + 2 \) is a quadratic function. Here, the quadratic polynomial comprises three terms: \( z^2 \), a linear term \(-2z\), and a constant term \( 2 \). This polynomial specifically tells us that the graph opens upwards, as \( a = 1 \) is positive.

The quadratic formula is a universal method for finding the zeros, or roots, of any quadratic function. It is expressed as:\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula allows us to solve quadratic equations when they cannot be easily factored.

To use the formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
• Plug these values into the formula.
• Simplify inside the square root first, then complete the calculations.

In our particular exercise, the given quadratic function is \( f(z) = z^2 - 2z + 2 \), with:\
- \( a = 1 \)
- \( b = -2 \)
- \( c = 2 \)
Substituting these into the quadratic formula, we calculate:\[ z = \frac{2 \pm \sqrt{(-2)^2 - 4 \times 1 \times 2}}{2 \times 1} \]
This results in a negative value under the square root (discriminant), indicating that the function does not have real roots. Instead, the roots are complex numbers, meaning the graph does not intersect the x-axis.

After finding the zeros of a quadratic function through the quadratic formula, the next step is to express the function as a product of linear factors. That means rewriting the quadratic polynomial using its roots.
The general form of expressing a quadratic equation \( ax^2 + bx + c \) in terms of its roots \( r_1 \) and \( r_2 \) is:\[ f(x) = a(x - r_1)(x - r_2) \]Each factor \( (x - r_i) \) corresponds to a zero \( r_i \).
In our example, the quadratic function \( f(z) = z^2 - 2z + 2 \) was determined to have complex roots rather than real ones, because the discriminant \( b^2 - 4ac \) was negative. This means that the expression in terms of linear factors will incorporate imaginary numbers, represented in conjugate pairs.
The roots \( z_1 \) and \( z_2 \) might be expressed as \( z = 1 \pm i \), leading to the factorization as \( (z - (1 + i))(z - (1 - i)) \). Polynomial linear factors are significant because they allow us to understand more about the structure and solutions of quadratic polynomials, including where (or "if") they cross the x-axis.