Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. h(x)=-(x+1)^3

Verified step by step guidance

1. Identify the basic function: The function given is a transformation of the basic cubic function \( f(x) = x^3 \).
2. Determine the transformations: The function \( h(x) = -(x+1)^3 \) involves a horizontal shift, a reflection, and a vertical stretch/compression.
3. Horizontal shift: The term \((x+1)\) indicates a shift to the left by 1 unit.
4. Reflection: The negative sign in front of \((x+1)^3\) indicates a reflection across the x-axis.
5. Graph the function: Start with the basic graph of \( f(x) = x^3 \), apply the left shift, and then reflect it across the x-axis.

Key Concepts

Function Graphing

Graphing a function involves plotting points on a coordinate plane that represent the output values of the function for given input values. Understanding how to interpret the function's equation and its transformations is crucial for accurately representing its behavior visually.

Transformations of Functions

Transformations refer to changes made to the graph of a function, including shifts, reflections, stretches, and compressions. In the function h(x)=-(x+1)^3, the negative sign indicates a reflection over the x-axis, while the (x+1) indicates a horizontal shift to the left by one unit.

Cubic Functions

Cubic functions are polynomial functions of degree three, characterized by their general form f(x) = ax^3 + bx^2 + cx + d. They typically exhibit an 'S' shaped curve and can have one or two turning points, which are important for understanding their overall shape and behavior when graphed.