Horizontal Stretch

Definition

A horizontal stretch is a transformation of a function where the input values are scaled or compressed along the x-axis, effectively widening or narrowing the graph of the function. This transformation affects the period and frequency of the function, and has important implications in the context of function transformations, trigonometric functions, and parabolas.

5 Must Know Facts For Your Next Test

1. A horizontal stretch of a function $f(x)$ is achieved by replacing $x$ with $a \cdot x$, where $a$ is a positive constant greater than 1, resulting in the transformed function $f(a \cdot x)$.
2. Horizontally stretching a function by a factor of $a$ will increase the period of the function by a factor of $a$, and decrease the frequency by a factor of $1/a$.
3. Horizontal stretches can be used to adjust the frequency and period of trigonometric functions, such as sine and cosine, to model real-world phenomena more accurately.
4. In the context of parabolas, a horizontal stretch will widen the parabola, changing the rate of change and affecting the vertex and focus of the parabolic function.
5. Horizontal stretches, along with other transformations, are essential in understanding the behavior and characteristics of various families of functions.

Review Questions

• Explain how a horizontal stretch affects the period and frequency of a function.
  • A horizontal stretch of a function $f(x)$ by a factor of $a$ (where $a > 1$) is achieved by replacing $x$ with $a \cdot x$, resulting in the transformed function $f(a \cdot x)$. This horizontal stretch increases the period of the function by a factor of $a$, and decreases the frequency by a factor of $1/a$. In other words, the function will repeat itself at a longer interval, but with fewer cycles occurring within a given interval.
• Describe the impact of a horizontal stretch on the graph of a parabolic function.
  • When a parabolic function $f(x) = ax^2 + bx + c$ undergoes a horizontal stretch by a factor of $a$, the resulting transformed function is $f(a \cdot x) = a(a \cdot x)^2 + b(a \cdot x) + c$. This horizontal stretch will widen the parabola, changing the rate of change and affecting the location of the vertex and focus of the parabolic function. Specifically, the vertex will be shifted horizontally by a factor of $1/a$, and the focus will be shifted horizontally by a factor of $1/(4a)$.
• Analyze how horizontal stretches can be used to model real-world phenomena involving trigonometric functions.
  • Horizontal stretches of trigonometric functions, such as sine and cosine, can be used to adjust the period and frequency of the functions to more accurately model real-world periodic phenomena. For example, the motion of a pendulum or the vibrations of a guitar string can be represented by a sine or cosine function, and a horizontal stretch can be applied to the function to match the observed period and frequency of the real-world system. By understanding how horizontal stretches affect the characteristics of trigonometric functions, researchers and engineers can develop more accurate mathematical models to describe and predict the behavior of complex periodic systems.

Related terms

Dilation: A dilation is a transformation that scales or resizes a function, either expanding or contracting the graph.

Period: The period of a function is the distance between two consecutive repeating points on the graph, representing the interval at which the function repeats itself.

Frequency: The frequency of a function is the number of complete cycles that occur within a given interval, which is inversely related to the period of the function.