Graph each function. ƒ(x) = 2∛(x+1)-2

Verified step by step guidance

1. Identify the type of function: The given function ƒ(x) = 2∛(x+1)-2 is a transformation of the basic cube root function, y = ∛x.
2. Determine the transformations applied: The '+1' inside the cube root shifts the graph left by 1 unit. The '2' coefficient outside the cube root vertically stretches the graph by a factor of 2. The '-2' outside the cube root shifts the graph downward by 2 units.
3. Plot key points: Start by plotting points for the basic cube root function y = ∛x, such as (-1, -1), (0, 0), and (1, 1). Apply the transformations to these points: shift left by 1, stretch vertically by 2, and shift down by 2.
4. Draw the graph: Using the transformed points, sketch the graph. The cube root function has an 'S' shape, passing through the transformed points. Ensure the graph approaches infinity as x goes to positive infinity and negative infinity as x goes to negative infinity.
5. Check for symmetry and asymptotes: The function does not have symmetry and does not have horizontal or vertical asymptotes. The graph should smoothly continue in both the positive and negative directions of the x-axis.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values). Understanding how to create a graph requires knowledge of the function's behavior, including its intercepts, asymptotes, and overall shape. This process helps in analyzing the function's characteristics and identifying key features.

Cube Root Function

The cube root function, represented as ƒ(x) = ∛x, is a type of radical function that returns the number which, when cubed, gives the input value. It is defined for all real numbers and has a characteristic shape that passes through the origin. The transformation of this function, such as shifting and scaling, affects its graph, which is essential for understanding the given function ƒ(x) = 2∛(x+1)-2.

Transformations of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the function ƒ(x) = 2∛(x+1)-2, the '+1' indicates a horizontal shift to the left by 1 unit, while the '-2' indicates a vertical shift downward by 2 units. Understanding these transformations is crucial for accurately graphing the function and predicting its behavior based on the original cube root function.