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. Start by graphing the parent function \( f(x) = \sqrt[3]{x} \). This is the cube root function, which has a characteristic S-shape. It passes through the origin (0, 0), is symmetric about the origin, and increases as x increases.
2. Next, analyze the transformation \( f(x) = \sqrt[3]{-x} \). The negative sign inside the cube root reflects the graph of \( \sqrt[3]{x} \) across the y-axis. This means that for every point (x, y) on the original graph, the new graph will have the point (-x, y).
3. Now, consider the transformation \( f(x) = \sqrt[3]{-x - 2} \). The \(-2\) inside the cube root shifts the graph to the left by 2 units. This means that every point on the graph of \( \sqrt[3]{-x} \) will move 2 units to the left.
4. Combine the transformations: Start with the parent function \( \sqrt[3]{x} \), reflect it across the y-axis to get \( \sqrt[3]{-x} \), and then shift the resulting graph 2 units to the left to obtain \( \sqrt[3]{-x - 2} \).
5. Finally, plot the transformed graph. Key points to include are the new origin (shifted to (-2, 0)) and other points that reflect the transformations. Ensure the graph maintains the S-shape characteristic of cube root functions.

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 applying transformations.

Graph Transformations

Graph transformations involve altering the position or shape of a function's graph through operations such as translations, reflections, and stretches. For instance, the function f(x) = ∛(-x-2) represents a horizontal shift and a reflection across the y-axis of the original cube root function. Mastery of these transformations allows for the accurate graphing of modified functions.

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of transformations that move the graph of a function along the x-axis and y-axis, respectively. A horizontal shift occurs when a constant is added or subtracted from the input variable, while a vertical shift involves adding or subtracting a constant from the function's output. In the case of ∛(-x-2), the graph shifts left by 2 units and reflects across the y-axis.