A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This type of transformation is often used to model real-world phenomena and can be applied to various basic classes of functions.
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A horizontal shift is represented by the addition or subtraction of a constant to the independent variable (x) of the function.
The direction of the horizontal shift depends on the sign of the constant: a positive constant shifts the graph to the left, while a negative constant shifts the graph to the right.
Horizontal shifts do not affect the range or domain of the function, but they can change the x-intercepts and other key features of the graph.
Horizontal shifts are commonly used to model changes in the input or independent variable of a function, such as changes in time, location, or other factors.
Horizontal shifts can be combined with other transformations, such as vertical shifts, stretches/compressions, and reflections, to create more complex transformations of functions.
Review Questions
Explain how a horizontal shift affects the graph of a function.
A horizontal shift of a function moves the graph left or right along the x-axis, without changing the shape or orientation of the graph. If the constant added to the independent variable (x) is positive, the graph shifts to the left; if the constant is negative, the graph shifts to the right. The range and domain of the function are not affected by a horizontal shift, but the x-intercepts and other key features of the graph may change.
Describe how a horizontal shift can be used to model real-world phenomena.
Horizontal shifts can be used to model changes in the input or independent variable of a function, such as changes in time, location, or other factors. For example, a function describing the position of an object over time can be horizontally shifted to represent the object starting at a different location. Similarly, a function describing the growth of a population can be horizontally shifted to represent the population starting at a different size or time.
Analyze how a horizontal shift can be combined with other transformations to create more complex function graphs.
Horizontal shifts can be combined with other transformations, such as vertical shifts, stretches/compressions, and reflections, to create more complex transformations of functions. For instance, a function could be horizontally shifted, then vertically shifted, and finally reflected across the x-axis. This combination of transformations would result in a graph that is both shifted and transformed in multiple ways, allowing for the modeling of more complex real-world phenomena. Understanding how to apply and combine these various transformations is crucial for understanding the behavior of functions in the context of 1.2 Basic Classes of Functions.
A vertical shift is a transformation that moves the graph of a function up or down along the y-axis, without changing the shape or orientation of the graph.
Stretch/Compression: A stretch or compression is a transformation that changes the scale of a function, either horizontally or vertically, without changing the overall shape of the graph.
A reflection is a transformation that flips the graph of a function across either the x-axis or the y-axis, creating a mirror image of the original graph.