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.