Question

Begin by graphing the standard cubic function, f(x) = x3. Then use transformations of this graph to graph the given function. g(x) = (x − 3)3

Accepted Answer

Start by recalling the standard cubic function f(x) = x³. This function has a characteristic S-shaped curve, passing through the origin (0, 0), with symmetry about the origin. The graph increases to the right and decreases to the left. Understand the transformation applied to f(x) = x³ to obtain g(x) = (x − 3)³. The term (x − 3) indicates a horizontal shift. Specifically, the graph of f(x) = x³ is shifted 3 units to the right. To graph g(x) = (x − 3)³, take each key point on the graph of f(x) = x³ (e.g., (-1, -1), (0, 0), (1, 1), etc.) and shift it 3 units to the right. For example, the point (0, 0) on f(x) becomes (3, 0) on g(x). Sketch the new graph by connecting the shifted points smoothly, maintaining the same S-shaped curve as the original cubic function. Ensure the graph still increases to the right and decreases to the left, with the inflection point now at (3, 0). Label the graph of g(x) = (x − 3)³ clearly, and verify that the transformation has been applied correctly by checking a few additional points. For example, if x = 4, g(4) = (4 − 3)³ = 1³ = 1, so the point (4, 1) should be on the graph.

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = (x − 3)³

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Cubic Functions

Graph Transformations

Horizontal Shifts