key term - Horizontal Stretch

Definition

A horizontal stretch is a transformation applied to a function that changes the rate of change or slope of the function along the x-axis. It affects the width or spread of the function's graph without altering its height or vertical position.

5 Must Know Facts For Your Next Test

1. A horizontal stretch is achieved by multiplying the independent variable (x) by a positive constant value greater than 1, which widens the graph of the function.
2. Horizontal stretches are the opposite of horizontal compressions, where the graph is narrowed by multiplying the x-value by a positive constant less than 1.
3. Horizontal stretches and compressions do not affect the y-intercept or the range of the function, but they do change the domain and the rate of change.
4. The magnitude of the horizontal stretch is determined by the value of the constant used to multiply the x-variable; larger values result in greater stretches.
5. Horizontal stretches can be combined with other transformations, such as vertical stretches, shifts, or reflections, to create more complex transformations of a function.

Review Questions

• Explain how a horizontal stretch affects the graph of a function.
  • A horizontal stretch of a function's graph widens or expands the graph along the x-axis, without changing its height or vertical position. This is achieved by multiplying the x-values of the function by a positive constant greater than 1. The larger the constant value, the greater the horizontal stretch, and the more spread out the graph will appear. Horizontal stretches do not affect the y-intercept or range of the function, but they do change the domain and rate of change.
• Compare and contrast the effects of a horizontal stretch and a horizontal compression on the graph of a function.
  • A horizontal stretch and a horizontal compression are inverse transformations of each other. A horizontal stretch, achieved by multiplying the x-values by a positive constant greater than 1, widens the graph of the function along the x-axis. In contrast, a horizontal compression, achieved by multiplying the x-values by a positive constant less than 1, narrows the graph of the function along the x-axis. Both transformations affect the domain and rate of change of the function, but they have opposite effects on the overall width or spread of the graph. The y-intercept and range of the function remain unchanged for both horizontal stretches and compressions.
• Describe how a horizontal stretch can be combined with other transformations to create more complex changes to the graph of a function.
  • Horizontal stretches can be combined with other transformations, such as vertical stretches, shifts, or reflections, to create more intricate changes to the graph of a function. For example, a horizontal stretch combined with a vertical stretch would result in a graph that is both wider and taller than the original parent function. Combining a horizontal stretch with a vertical shift would shift the entire stretched graph up or down the y-axis. Furthermore, applying a horizontal stretch and then reflecting the graph about the x-axis or y-axis would create a mirrored, wider version of the original function. These combinations of transformations allow for a wide range of modifications to the shape, size, and position of a function's graph.

Related terms

Transformation: A change applied to a function that alters the appearance or behavior of its graph, such as shifting, stretching, or reflecting.

Dilation: A type of transformation that uniformly scales or resizes the size of a function's graph, either horizontally, vertically, or both.

Parent Function: The original, unmodified function that serves as the basis for transformations, such as stretches or shifts.