Begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛(x-2)

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Start by graphing the parent function f(x) = ∛x. This is the cube root function, which has a characteristic shape: it passes through the origin (0, 0), increases slowly for positive x, and decreases slowly for negative x. The graph is symmetric about the origin, meaning it has odd symmetry.

Understand the transformation applied to the parent function. The given function is g(x) = ∛(x - 2). The term (x - 2) inside the cube root indicates a horizontal shift. Specifically, the graph of f(x) = ∛x is shifted 2 units to the right.

To apply the horizontal shift, take key points from the graph of f(x) = ∛x, such as (-1, -1), (0, 0), and (1, 1), and adjust their x-coordinates by adding 2. For example, (-1, -1) becomes (1, -1), (0, 0) becomes (2, 0), and (1, 1) becomes (3, 1).

Plot the transformed points on the graph and sketch the curve, ensuring it retains the same shape as the parent function but is shifted 2 units to the right.

Label the graph of g(x) = ∛(x - 2) clearly, and verify that the transformation has been applied correctly by checking additional points if necessary.

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주요 개념

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Cube Root Function

The cube root function, denoted as f(x) = ∛x, is a type of radical function that returns the number which, when cubed, gives the input x. This function is defined for all real numbers and has a characteristic S-shaped curve that passes through the origin (0,0). Understanding its basic shape and properties is essential for graphing transformations.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the transformation g(x) = ∛(x-2) represents a horizontal shift of the cube root function f(x) = ∛x to the right by 2 units. Recognizing how these transformations affect the graph is crucial for accurately sketching the new function.

Horizontal Shifts

Horizontal shifts occur when a function is modified by adding or subtracting a constant to the input variable. For g(x) = ∛(x-2), the '-2' indicates a shift to the right, meaning every point on the original graph of f(x) is moved 2 units to the right. This concept is vital for understanding how the position of the graph changes without altering its shape.