Here are the essential concepts you must grasp in order to answer the question correctly.
Cubic Functions
A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax³ + bx² + cx + d. The standard cubic function, f(x) = x³, has a characteristic S-shaped curve that passes through the origin and extends infinitely in both directions. Understanding the basic shape and properties of cubic functions is essential for analyzing transformations.
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Graph Transformations
Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For the function g(x) = x³ - 3, the '-3' indicates a vertical shift downward by three units. Recognizing how these transformations affect the original graph of f(x) = x³ is crucial for accurately graphing the new function.
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Vertical Shifts
A vertical shift occurs when a constant is added to or subtracted from a function, resulting in the entire graph moving up or down. In the case of g(x) = x³ - 3, the graph of f(x) = x³ is shifted down by three units. This concept is fundamental in understanding how the position of the graph changes without altering its shape.
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