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Stretching and Shrinking
Stretching and shrinking refer to transformations that alter how compact a function looks in the x or y direction.
Learning Objective

Operate on functions so that they stretch or shrink
Key Points

When by either f(x) or x is multiplied by a number, functions can "stretch" or "shrink" vertically or horizontally, respectively, when graphed.

In general, a vertical stretch is given by the equation y = pf(x). If p>1, the graph stretches upward and downward. If p<1, the graph shrinks.

In general, a horizontal stretch is given by the equation y = f(x/q). If q>1, the graph shrinks horizontally, becoming more compact. If q<1, the graph stretches horizontally.
Term
Full Text
In algebra, equations can be stretched horizontally or vertically along an axis by multiplying either x or y by a number, respectively. By multiplying f(x) by a number greater than one, all the y values of an equation increase. This leads to a "stretched" appearance in the vertical direction. If f(x) is multiplied by a value less than one, all the y values of the equation decrease, leading to a "shrunken" appearance in the vertical direction. Alternatively, if only x is multiplied, the graph stretches or shrinks in the horizontal direction.
For examples, we will use the basic trignometric function f(x) = sin(x), which is black in the two graphs in . Stretches can be a bit confusing with linear or quadratic functions, but they are much more straight forward with the sine function. The red function in has been stretched (dilated) vertically by a factor of 3 and follows the equation:
In general a vertical stretch is given by the equation:
If p is larger than 1, the function gets "taller. " If p is smaller than 1, the function gets "shorter. "
The blue function in has been been stretched horizontally by a factor of 3 and has the equation:
In general, a horizontal stretch is given by the equation:
In the example above, q = 1/3. When q is larger than 1, the function will get "longer" and when q is smaller than 1, the function will "squish".
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Key Term Reference
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 factor
 Appears in this related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Average Value of a Function
 graph
 Appears in this related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Equations
 linear
 Appears in this related concepts: Exponential Growth and Decay, SecondOrder Linear Equations, and Linear Approximation
 quadratic function
 Appears in this related concepts: Quadratic Equations and Quadratic Functions, Reducing Equations to a Quadratic, and Applications and ProblemSolving
 transformation
 Appears in this related concepts: Horizontal Gene Transfer, Reproduction, and Genomic similarities between distant species
Sources
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Cite This Source
Source: Boundless. “Stretching and Shrinking.” Boundless Algebra. Boundless, 29 Dec. 2014. Retrieved 26 May. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/graphsfunctionsandmodels2/transformations21/stretchingandshrinking1201623/