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

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1. <Step 1: Understand the parent function.> The parent function here is \( f(x) = \sqrt[3]{x} \). This is the cube root function, which has a characteristic shape that passes through the origin (0,0) and is symmetric about the origin.
2. <Step 2: Graph the parent function.> Plot the graph of \( f(x) = \sqrt[3]{x} \). It will have a gentle S-shape, increasing from left to right, passing through the origin, and extending infinitely in both directions.
3. <Step 3: Identify the transformation.> The given function is \( g(x) = \sqrt[3]{x} + 2 \). This represents a vertical shift of the parent function. The '+2' indicates that the entire graph of \( f(x) = \sqrt[3]{x} \) will be shifted upwards by 2 units.
4. <Step 4: Apply the transformation.> Take each point on the graph of \( f(x) = \sqrt[3]{x} \) and move it up 2 units. For example, the point (0,0) on \( f(x) \) will move to (0,2) on \( g(x) \).
5. <Step 5: Graph the transformed function.> Draw the new graph of \( g(x) = \sqrt[3]{x} + 2 \) using the points obtained from the transformation. The graph will maintain the same shape as \( f(x) = \sqrt[3]{x} \) but will be shifted upwards by 2 units.

Key Concepts

Cube Root Function

The cube root function, f(x) = ∛x, is a fundamental mathematical function that returns the number whose cube is x. It 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 applied to the cube root function to obtain g(x) = ∛(x + 2) is a horizontal shift to the left by 2 units. Recognizing how these transformations affect the graph is crucial for accurately sketching the new function.

Function Notation

Function notation, such as f(x) and g(x), is a way to represent functions and their outputs. It allows for clear communication of the relationship between input values (x) and their corresponding outputs (f(x) or g(x)). Understanding function notation is vital for interpreting and manipulating functions correctly in algebra.