Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = -x³

Verified step by step guidance

1. Start by identifying the standard cubic function, f(x) = x³. This function has a characteristic S-shaped curve that passes through the origin (0, 0), with the left side decreasing and the right side increasing symmetrically.
2. Next, analyze the given function, h(x) = -x³. The negative sign in front of x³ indicates a reflection of the graph of f(x) = x³ across the x-axis. This means that the portions of the graph that were above the x-axis will now be below it, and vice versa.
3. To graph h(x) = -x³, begin by plotting key points from the standard cubic function, such as (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8). Reflect these points across the x-axis to get the corresponding points for h(x). For example, (1, 1) becomes (1, -1), and (-2, -8) becomes (-2, 8).
4. Draw a smooth curve through the reflected points, ensuring that the graph maintains the same general S-shape as the standard cubic function but flipped vertically.
5. Finally, verify the transformation by checking additional points or using a graphing tool to confirm that the graph of h(x) = -x³ is indeed the reflection of f(x) = x³ across the x-axis.

Key Concepts

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax³ + bx² + cx + d. The standard cubic function, f(x) = x³, has a characteristic S-shaped curve that passes through the origin and extends infinitely in both directions. Understanding the basic shape and properties of cubic functions is essential for analyzing transformations.

Graph Transformations

Graph transformations involve altering the position or shape of a graph through various operations, such as translations, reflections, stretches, and compressions. For example, the function h(x) = -x³ represents a reflection of the standard cubic function across the x-axis. Recognizing how these transformations affect the graph is crucial for accurately sketching the transformed function.

Reflection Across the X-Axis

Reflection across the x-axis occurs when a graph is flipped over the x-axis, resulting in a change of sign for the y-values of the function. In the case of h(x) = -x³, every point (x, y) on the graph of f(x) = x³ is transformed to (x, -y). This concept is vital for understanding how the original cubic function's shape is altered when applying a negative coefficient.