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

Verified step by step guidance

1. Step 1: Begin by understanding the parent function f(x) = ∛x. The cube root function is symmetric about the origin and passes through points such as (0, 0), (1, 1), (-1, -1), (8, 2), and (-8, -2). Graph this parent function as the base graph.
2. Step 2: Analyze the transformation inside the cube root function. The term (x + 2) indicates a horizontal shift. Since it is (x + 2), the graph of the parent function will shift 2 units to the left.
3. Step 3: Next, consider the negative sign in front of the cube root function, -∛(x + 2). This negative sign reflects the graph across the x-axis. Every y-value of the graph will be multiplied by -1, flipping the graph vertically.
4. Step 4: Combine the transformations. First, shift the graph of f(x) = ∛x two units to the left, and then reflect the resulting graph across the x-axis.
5. Step 5: Plot the transformed graph by applying the transformations to key points of the parent function. For example, the point (0, 0) remains unchanged, (1, 1) becomes (-1, -1), and (-1, -1) becomes (-3, 1). Continue this process for other key points to complete the graph.

Key Concepts

Cube Root Function

The cube root function, denoted as f(x) = ∛x, is a type of radical 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 applying transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For the function -∛(x+2), the graph of f(x) = ∛x is first shifted left by 2 units due to the (x+2) term, and then reflected across the x-axis because of the negative sign. Mastery of these transformations allows for accurate graphing of modified functions.

Reflection Across the X-Axis

Reflection across the x-axis is a transformation that flips the graph of a function over the x-axis. This means that for every point (x, y) on the original graph, the reflected point will be (x, -y). In the context of the function -∛(x+2), this reflection changes the sign of the output values, resulting in a graph that is inverted compared to the original cube root function.