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In the world of mathematics, particularly when dealing with function transformations, the term horizontal shift relates to moving the graph of a function left or right along the horizontal axis. Imagine you have a point on the graph that's initially at the origin (0,0). If we apply a horizontal shift of 3 units to the right, this point would simply move to (3,0).

When we express function transformations algebraically, a horizontal shift to the right by 'k' units is indicated by \(x - k\) inside the function's rule. Conversely, a shift to the left is indicated by \(x + k\). It's essential to remember that despite the intuitive notion that moving right involves a positive number, within function notation, it’s represented as a subtraction.

Delving deeper into the concept of transformations, we aren't limited to just shifting; transformations of functions can include stretching, compressing, reflecting, and translating the graph of a function. A function's transformation is an alteration that produces a new function with a graph that is a modification of the original function's graph. The basic transformations include:

• Translations: Movement of the graph up, down, left, or right
• Reflections: Flipping the graph across an axis
• Rescaling: Stretching or compressing the graph vertically or horizontally

Understanding these transformations allows us to take a basic function and adjust it to describe more complex situations as represented by their respective transformations.

Cubic functions are a family of polynomial functions characterized by their distinctive S-shaped curves, which we see in the graph of \(f(x) = x^3\). These functions have the general form of \(f(x) = ax^3 + bx^2 + cx + d\), where 'a', 'b', 'c', and 'd' are constants with 'a' not equal to zero. A cubic function can exhibit one, two, or three real roots, one maximum and one minimum point, or what we call 'turning points'.

Cubic functions' behaviors differ vastly from linear and quadratic functions, with their inflection point causing a change in the concavity of the curve. This point, along with the rate of increase or decrease of the function, provides diverse applications in real-world problem solving and mathematical analysis.

Function notation is a way of representing functions that is both compact and provides an easy reference for evaluation, comparison, and transformation. For example, when we write \(f(x)\), we're identifying a function named 'f' with 'x' as the variable. This notation is incredibly powerful because it allows us to succinctly represent the application of a function on a specific input value.

The utilization of function notation becomes clearer when we manipulate functions, such as in transformations. To indicate that we are working with a recycled version of the original function, after some alteration, we still use the same function name but adjust the input value to reflect the change. For instance, a horizontal shift of 13 units to the right for the function \(f(x) = x^3\) would be denoted as \(f(x - 13)\), signifying that we evaluate the original cubic function at \(x - 13\), not just 'x'.