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Quadratic functions are a type of polynomial function characterized by the highest power of the variable being 2. They are represented in the general form \( ax^2 + bx + c \). The graph of a quadratic function is completely described by a parabola.
Quadratic curves can either open upwards or downwards depending on the sign of the coefficient of the \(x^2\) term (\(a\)). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. These functions are often used to model paths or trajectories in physics.
Quadratic functions have several interesting properties:

• Their graphs are symmetrical around a central axis called the axis of symmetry.
• They have a single turning point known as the vertex.
• The roots or x-intercepts can be found using methods like factoring, completing the square, or using the quadratic formula.

Understanding the behavior of quadratic functions is crucial, especially when it comes to transformations and modifications like in our exercise.

Absolute value functions involve mathematical expressions that return the distance of a number from zero on a number line, essentially stripping any negative sign from the value. These functions are written as \(|x|\) and always return non-negative results.
Absolute value can be thought of as a piecewise function:

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

In our exercise, the absolute value function \(|f(x)|\) plays a crucial role in transforming \(f(x)\). It ensures that any negative output of \(f(x)\) becomes positive, which effectively transforms part of the graph that was below the x-axis to move onto the x-axis.
This characteristic is vital when mixing with other functions, as it can dramatically change the outcome of a mathematical transformation.

The vertex form of a quadratic function is particularly useful because it provides immediate information about the vertex of a parabola, which is the point at which the direction changes from downward to upward or vice versa.
The vertex form is given by \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. The advantage of this is that it directly reveals the minimum or maximum value of the quadratic function, depending on whether the parabola opens upwards or downwards.
To convert a standard quadratic equation to vertex form involves completing the square. This transformation simplifies understanding of the graph's attributes like its vertex and helps in graphical translations and scaling, as evident when modifying \(f(x)\) in our exercise.
The vertex form sheds light on the minimum value of \(f(x)\), allowing us to effectively determine its range, especially when modifications like absolute value alterations are involved.

The range of a function refers to the complete set of all possible output values after substituting the domain's values into the function. Understanding the range gives insight into the vertical span of a graph.
For the quadratic function from our exercise, \(f(x) = x^2 - 6x + 5\) can be written in vertex form \((x-3)^2 - 4\), revealing its vertex at \((3, -4)\). Because this quadratic opens upwards, the minimum y-value is \(-4\) and the parabola continues to infinity upward, setting the range as \([-4, \infty)\).
With function transformations, understanding how these affect the range is crucial. In the case of \(g(x) = \frac{1}{2}[f(x) + |f(x)|]\), introducing the absolute value component effectively clips all values below zero to zero itself, thus altering the range. The modifications due to absolute values can transform expected outcomes, as shown where parts of \(f(x)\) dip below the x-axis.