In Exercises 107-118, 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)

Verified step by step guidance

1. Start by graphing the basic cube root function, \( f(x) = \sqrt[3]{x} \). This graph passes through the origin (0,0) and is symmetric about the origin, with a gentle S-shape.
2. Identify the transformation needed to graph \( g(x) = \sqrt[3]{x-2} \). Notice that the expression inside the cube root, \( x-2 \), indicates a horizontal shift.
3. Understand that \( x-2 \) represents a shift to the right by 2 units. This is because the function \( \sqrt[3]{x} \) is shifted horizontally by the value that makes the expression inside the cube root zero.
4. Apply the horizontal shift to the graph of \( f(x) = \sqrt[3]{x} \). Move every point on the graph of \( f(x) \) two units to the right to obtain the graph of \( g(x) = \sqrt[3]{x-2} \).
5. Verify the transformation by checking key points. For example, the point (0,0) on \( f(x) \) becomes (2,0) on \( g(x) \), and the point (1,1) on \( f(x) \) becomes (3,1) on \( g(x) \).

Key Concepts

Cube Root Function

The cube root function, denoted as f(x) = ∛x, is a fundamental mathematical function that returns the number which, when cubed, gives the input value 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 and transforming it.

Graph Transformations

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

Horizontal Shifts

A horizontal shift occurs when a function is modified by adding or subtracting a constant from the input variable. For g(x) = ∛(x-2), the '-2' indicates a shift of the graph of f(x) = ∛x to the right by 2 units. This concept is vital for understanding how the position of the graph changes without altering its shape.