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Cubic functions are polynomial functions given by the general form \( f(x) = ax^3 + bx^2 + cx + d \). A cubic function usually has its highest degree as 3 and is thus called a third-degree polynomial. These functions are characterized by their distinctive shape, which often includes one or two bends called inflection points. For the function \( P(x) = x^3 \), it is a simple cubic function that passes through the origin (0,0) and is symmetric about this point. It increases steeply where \( x > 0 \) and decreases where \( x < 0 \). This simple cubic graph has no turning points as its smooth curve transitions smoothly from negative infinity to positive infinity through the origin.

Graph transformations involve systematically altering a function's appearance on a coordinate plane. The basic transformations include shifting, reflecting, stretching, and compressing the graph. When we alter the function \( P(x) = c x^3 \) by changing \( c \), we are applying vertical transformations. These affect how the graph moves along the y-axis without changing its shape completely.

• A positive \( c \) keeps the graph in the same direction (increases), while a negative \( c \) would reflect it.
• When \( c \) is more than 1, the graph stretches, and it compresses when between 0 and 1.

This specific transformation retains the characteristic shape of a cubic function but may make it appear steeper or flatter depending on the value of \( c \).

Vertical stretching and compression directly influence the steepness of a graph. Vertical Stretching: This occurs when the constant \( c \) in \( P(x) = c x^3 \) is greater than 1. For instance, \( c = 2 \) and \( c = 5 \) scale the graph vertically, making the slope steeper and the points rise or drop more sharply than the basic curve \( P(x) = x^3 \). With stretching, the output or \( y \)-values become a multiple of the original function.Vertical Compression: This effect takes place when \( 0 < c < 1 \). As seen with \( c = \frac{1}{2} \), it compresses the cubic graph, reducing its steepness. The effect is such that for any chosen \( x \)-value, the associated \( y \) is only a fraction of that in the original \( P(x) = x^3 \). This reduced slope is akin to flattening the curve, expanding its spread along the x-axis without altering its base form.