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Concept Version 11
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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

  • 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 $x$ or $y$ by a constant, respectively.  

Vertical Scaling 

First, let's talk about vertical scaling.  Multiplying the entire function $f(x)$ by a constant greater than one causes all the $y$ values of an equation to increase. This leads to a "stretched" appearance in the vertical direction. If the function $f(x)$ is multiplied by a value less than one, all the $y$ values of the equation will decrease, leading to a "shrunken" appearance in the vertical direction.  In general, the equation for vertical scaling is:

$\displaystyle y = bf(x)$

where $f(x)$ is some function and $b$ is an arbitrary constant.  If $b$ is greater than one the function will undergo vertical stretching, and if $b$ is less than one the function will undergo vertical shrinking.  

As an example, consider the initial sinusoidal function presented below:

$\displaystyle y = \sin(x)$  

If we want to vertically stretch the function by a factor of three, then the new function becomes:

$\displaystyle \begin{aligned} y &= 3f(x) \\ &= 3\sin(x) \end{aligned}$

Vertical scaling

The function $y=\sin(x)$ is stretched by a factor of three in the $y$ direction.

Horizontal Scaling 

Now lets analyze horizontal scaling.  Multiplying the independent variable $x$ by a constant greater than one causes all the $x$ values of an equation to increase. This leads to a "shrunken" appearance in the horizontal direction. If the independent variable $x$ is multiplied by a value less than one, all the x values of the equation will decrease, leading to a "stretched" appearance in the horizontal direction.  In general, the equation for horizontal scaling is:

$\displaystyle y = f(cx)$

where $f(x)$ is some function and $c$ is an arbitrary constant.  If $c$ is greater than one the function will undergo horizontal shrinking, and if $c$ is less than one the function will undergo horizontal stretching.  

As an example, consider again the initial sinusoidal function:

$\displaystyle y = \sin(x)$

If we want to induce horizontal shrinking, the new function becomes:

$\displaystyle \begin{aligned} y &= f(3x)\\ &= \sin(3x) \end{aligned}$

Horizontal scaling

The function $y=\sin(x)$ is shrunk by a factor of three in the $x$ direction.

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