Graphing Tools: Vertical and Horizontal Scaling (Part 2)
(This page is Part 2. Click here for Part 1.)
Ideas Regarding Horizontal Scaling (Stretching/Shrinking)

Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,.$
Points on the graph of $\,y=f(3x)\,$ are of the form $\,\bigl(x,f(3x)\bigr)\,.$

How can we locate these desired points $\,\bigl(x,f(3x)\bigr)\,$?
First, go to the point $\,\color{red}{\bigl(3x\,,\,f(3x)\bigr)}\,$ on the graph of $\,\color{red}{y=f(x)\,.}$ This point has the $y$value that we want, but it has the wrong $x$value. The $x$value of this point is $\,3x\,,$ but the desired $x$value is just $\,x\,.$
Thus, the current $\,\color{purple}{x}$value must be divided by $\,\color{purple}{3}\,$; the $\,\color{purple}{y}$value remains the same. This gives the desired point $\,\color{green}{\bigl(x,f(3x)\bigr)}\,.$
Thus, the graph of $\,y=f(3x)\,$ is the same as the graph of $\,y=f(x)\,,$ except that the $x$values have been divided by $\,3\,$ (not multiplied by $\,3\,,$ which you might expect).
Notice that dividing the $x$values by $\,3\,$ moves them closer to the $y$axis; this is called a horizontal shrink.

Transformations involving $\,x\,$ do NOT work the way you would expect them to work! They are counterintuitive—they are against your intuition.

Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$ and asked about the graph of $\,y=f(3x)\,$:
interpretation of new equation:
$$ \cssId{s21}{y = f( \overset{\text{replace $x$ by $3x$}}{\overbrace{ \ \ 3x\ \ }}} ) $$
Replacing every $\,x\,$ by $\,3x\,$ in an equation causes the $x$values in the graph to be DIVIDED by $\,3\,.$
Summary of Horizontal Scaling
Let $\,k\gt 1\,.$ Start with the equation $\,y=f(x)\,.$ Replace every $\,x\,$ by $\,kx\,$ to give the new equation $\,y=f(kx)\,.$
This causes the $x$values on the graph to be DIVIDED by $\,k\,,$ which moves the points closer to the $y$axis. This is called a horizontal shrink.
A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(\frac{a}{k},b)\,$ on the graph of $\,y=f(kx)\,.$
Additionally: Let $\,k\gt 1\,.$ Start with the equation $\,y=f(x)\,.$ Replace every $\,x\,$ by $\,\frac{x}{k}\,$ to give the new equation $\,y=f(\frac{x}{k})\,.$
This causes the $x$values on the graph to be MULTIPLIED by $\,k\,,$ which moves the points farther away from the $y$axis. This is called a horizontal stretch.
A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(ka,b)\,$ on the graph of $\,y=f(\frac{x}{k})\,.$
This transformation type is formally called horizontal scaling (stretching/shrinking).
Different Words Used to Talk About Transformations Involving $\,y\,$ and $\,x\,$
Notice that different words are used when talking about transformations involving $\,y\,,$ and transformations involving $\,x\,.$
For transformations involving $\,y\,$ (that is, transformations that change the $y$values of the points), we say:
DO THIS to the previous $\,y$value.
For transformations involving $\,x\,$ (that is, transformations that change the $x$values of the points), we say:
REPLACE the previous $x$values by $\ldots$
Make Sure You See The Difference!
Vertical Scaling:
going from
$\,y=f(x)\,$
to
$\,y = kf(x)\,$ for $\,k\gt 0$
Horizontal Scaling:
going from
$\,y = f(x)\,$
to
$\,y = f(kx)\,$ for $\,k\gt 0$
Make sure you see the difference between (say) $\,y = 3f(x)\,$ and $\,y = f(3x)\,$!
In the case of $\,y = 3f(x)\,,$ the $\,3\,$ is ‘on the outside’; we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then multiplying by $\,3\,.$ This is a vertical stretch.
In the case of $\,y = f(3x)\,,$ the $\,3\,$ is ‘on the inside’; we're multiplying $\,x\,$ by $\,3\,$ before dropping it into the $\,f\,$ box. This is a horizontal shrink.