The Cubic Parent Function: Properties And Transformations For Graphing And Building Other Functions.

Cubic Parent Function

x →∞, f(x) →∞x → -∞, f(x) → -∞

The cubic parent function is a polynomial function of degree three, which is defined by the equation f(x) = x^3. It is called a parent function because it serves as a basic building block for other cubic functions, which can be obtained by applying transformations to the parent function.

The graph of the cubic parent function is a simple, smooth curve that passes through the origin, with a steep rise on the right side and a steep fall on the left side. The function is odd, which means that it has rotational symmetry about the origin and is symmetric with respect to the x-axis.

The domain of the cubic parent function is the set of all real numbers, while the range is also the set of all real numbers. The function is always increasing on the interval (-∞, 0], and always decreasing on the interval [0, ∞). The inflection point of the function is at (0, 0), where it changes from concave upward to concave downward.

The cubic parent function can be transformed by applying different types of transformations to the parent function. For example, adding a constant term to the function shifts the graph vertically; multiplying the function by a constant stretches or compresses the graph vertically; and adding or subtracting a term within the parenthesis shifts the graph horizontally.

Overall, the cubic parent function serves as an important building block for many other functions in mathematics and is a crucial concept to understand when studying graphs and functions.

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