Graph each function. ƒ(x) = 2∛(x+1)-2

Verified step by step guidance

1. Identify the base function to graph, which is the cube root function: \(f(x) = \sqrt[3]{x}\). This function has a characteristic S-shaped curve passing through the origin (0,0).
2. Recognize the transformations applied to the base function. The function given is \(f(x) = 2\sqrt[3]{x+1} - 2\). Here, \(x+1\) indicates a horizontal shift, the coefficient 2 outside the cube root indicates a vertical stretch, and the \(-2\) indicates a vertical shift downward.
3. Apply the horizontal shift by replacing \(x\) with \(x+1\). This shifts the graph of \(\sqrt[3]{x}\) one unit to the left. So, the new 'center' point moves from (0,0) to (-1,0).
4. Apply the vertical stretch by multiplying the cube root by 2. This makes the graph steeper, stretching it vertically by a factor of 2.
5. Apply the vertical shift by subtracting 2 from the entire function, moving the graph down by 2 units. The new center point after all transformations is at (-1, -2). Plot this point and sketch the transformed cube root curve accordingly.

Key Concepts

Cube Root Function

The cube root function, denoted as ∛x, is the inverse of cubing a number. It produces real outputs for all real inputs and has an S-shaped curve passing through the origin. Understanding its basic shape helps in graphing transformations applied to it.

Function Transformations

Function transformations include shifts, stretches, and reflections applied to the parent function. In ƒ(x) = 2∛(x+1) - 2, the (x+1) inside the root shifts the graph left by 1 unit, the coefficient 2 vertically stretches it, and the -2 shifts it down by 2 units.

Graphing Techniques

Graphing involves plotting key points and understanding the shape of the function. For cube root functions, select values of x, compute corresponding y-values, and apply transformations. This step-by-step approach ensures an accurate sketch of the function.