Stretching and Shrinking
Stretching and shrinking refer to transformations that alter how compact a function looks in the
Learning Objective

Manipulate 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
$y=bf(x)$ . If$b>1$ , the graph stretches with respect to the$y$ axis, or vertically. If$b<1$ , the graph shrinks with respect to the$y$ axis.  In general, a horizontal stretch is given by the equation
$y = f(cx)$ . If$c>1$ , the graph shrinks with respect to the$x$ axis, or horizontally. If$c<1$ , the graph stretches with respect to the$x$ axis.
Term

scaling
A transformation that changes the size and/or shape of the graph of the function.
Full Text
In algebra, equations can undergo scaling, meaning they can be stretched horizontally or vertically along an axis. This is accomplished by multiplying either
Vertical Scaling
First, let's talk about vertical scaling. Multiplying the entire function
where
As an example, consider the initial sinusoidal function presented below:
If we want to vertically stretch the function by a factor of three, then the new function becomes:
Vertical scaling
The function
Horizontal Scaling
Now lets analyze horizontal scaling.
Multiplying the independent variable
where
As an example, consider again the initial sinusoidal function:
If we want to induce horizontal shrinking, the new function becomes:
Horizontal scaling
The function
Key Term Reference
 Scaling
 Appears in these related concepts: Types of Conic Sections, Introduction to Ellipses, and Transformations of Functions
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 direction
 Appears in these related concepts: Parallel and Perpendicular Lines, Slope, and Parabolas As Conic Sections
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 factor
 Appears in these related concepts: Rational Algebraic Expressions, Finding Factors of Polynomials, and Factors
 function
 Appears in these related concepts: Functions and Their Notation, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphing Equations, Graphical Representations of Functions, and Reading Points on a Graph
 independent variable
 Appears in these related concepts: Converting between Exponential and Logarithmic Equations, The Cartesian System, and What is a Quadratic Function?
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
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