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 base function to be graphed. Here, the base function is the cubic function \(f(x) = x^3\), which has a characteristic S-shaped curve passing through the origin (0,0).
2. Recognize the transformation inside the function \(h(x) = -(x+1)^3\). The term \((x+1)\) indicates a horizontal shift of the graph of \(f(x) = x^3\) to the left by 1 unit.
3. Note the negative sign in front of the cubic term. This reflects the graph of \(f(x) = x^3\) across the x-axis, flipping it upside down.
4. Combine the transformations: start with the graph of \(y = x^3\), shift it left by 1 unit to get \(y = (x+1)^3\), then reflect it across the x-axis to get \(y = -(x+1)^3\).
5. Plot key points to help sketch the graph: for example, find \(h(-2)\), \(h(-1)\), and \(h(0)\) by substituting these x-values into \(h(x) = -(x+1)^3\), then plot these points and draw a smooth curve through them reflecting the cubic shape and transformations.

Key Concepts

Graphing Cubic Functions

Cubic functions have the general form f(x) = ax^3 + bx^2 + cx + d and produce S-shaped curves. Understanding the shape and behavior of the basic cubic function y = x^3 helps in graphing transformations like shifts and reflections.

Transformations of Functions

Transformations include shifts, reflections, stretches, and compressions. For h(x) = -(x+1)^3, the '+1' inside the function shifts the graph left by 1 unit, and the negative sign reflects it across the x-axis.

Plotting Key Points and Using Symmetry

To graph accurately, identify key points such as the inflection point and points around it. Cubic functions are symmetric about their inflection point, which aids in plotting the curve smoothly.