Graph each function using the techniques of shifting,...

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^{2}$ ) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. $$ h(x)=-x^{3}+2 $$

Key Concepts

Domain and Range

Determining the domain and range is essential to understanding the behavior of a function. The domain represents all possible input values (x-values) for which the function is defined, while the range represents all possible output values (y-values) the function can produce. Transformations, like reflections and shifts, can affect the range while the domain might remain unchanged, depending on the function.

Reflection

Reflection is a transformation that 'flips' the graph over a specific axis. When a function is multiplied by -1, it reflects over the x-axis. This transformation changes the sign of all y-values for the points on the graph, resulting in a mirror-image effect about the horizontal axis.

Vertical Shift

A vertical shift moves the entire graph up or down on the coordinate plane. This is achieved by adding or subtracting a constant from the function. For example, adding 2 to a function results in every y-coordinate of the graph being increased by 2, shifting the graph upward without altering its shape.

Basic Function

This is the starting point for graph transformations, where a well-known function (such as the cubic function y = x^3) is used as a model. Understanding the basic function involves knowing its general shape, key points (like the origin for cubic functions), and inherent properties such as symmetry or inflection points, which serve as a foundation for further modifications.

Graph Transformations

Graph transformations involve altering the basic function by applying operations such as reflection, stretching, compressing, and shifting. These operations change the position, orientation, or scale of the graph without changing its fundamental behavior, allowing one to construct more complex functions from simpler ones.

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