Cubic Parent Function
x →∞, f(x) →∞x → -∞, f(x) → -∞
The cubic parent function is given by f(x) = x^3. This function is an example of a polynomial function of degree 3. It has a characteristic shape known as the cubic curve, which resembles an S-shaped curve.
The domain and range of the cubic parent function are both real numbers, (-∞, ∞). The function passes through the origin (0,0) and is symmetric about the origin.
The cubic function has two important features: it has a local minimum or maximum at the origin, and it increases without bound as x approaches infinity and decreases without bound as x approaches negative infinity.
The first derivative of the cubic parent function is f’(x) = 3x^2, which is always positive except at x = 0, where it is equal to 0. This indicates that the function is increasing on (-∞, 0) and (0, ∞) and has a local minimum or maximum at x = 0.
The second derivative of the cubic parent function is f’’(x) = 6x, which is positive for x > 0 and negative for x < 0. This confirms that the function has a local minimum at x = 0. The cubic function can be transformed using various operations, such as translation, reflection, stretching, and compressing. These transformations change the shape and position of the cubic curve, but the basic characteristics of the function remain the same. Overall, the cubic parent function is a fundamental function in mathematics, physics, and engineering, as it describes many natural phenomena and processes. It serves as a building block for more complex functions and is widely used in modeling and analysis of data.
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