Horizontal Shift

Definition

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.

5 Must Know Facts For Your Next Test

1. A horizontal shift is represented by the addition or subtraction of a constant to the independent variable (x) of the function.
2. 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.
3. 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.
4. 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.
5. 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.

Related terms

Vertical Shift: 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.

Reflection: 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.