Thanks for your support!

 

Here are ideas that are needed to understand graphical transformations.

• A function is a rule:
  it takes an input, and gives a unique output.
• If $\,x\,$ is the input to a function $\,f\,$,
  then the unique output is called $\,f(x)\,$ (which is read as ‘$\,f\,$ of $\,x\,$’).
• The graph of a function is a picture of all of its (input,output) pairs.
  We put the inputs along the horizontal axis (the $\,x\,$-axis),
  and the outputs along the vertical axis (the $\,y\,$-axis).
• Thus, the graph of a function $\,f\,$ is a picture of all points of the form $\,\bigl(x, \overset{\text{y-value}}{\overbrace{ f(x)}} \bigr) \,$.
  Here, $\,x\,$ is the input, and $\,f(x)\,$ is the corresponding output.
• The equation $\,y=f(x)\,$ is an equation in two variables, $\,x\,$ and $\,y\,$.
  A solution is a choice for $\,x\,$ and a choice for $\,y\,$ that makes the equation true.
  Of course, in order for this equation to be true, $\,y\,$ must equal $\,f(x)\,$.
  Thus, solutions to the equation $\,y=f(x)\,$ are points of the form $\,\bigl(x, \overset{\text{y-value}}{\overbrace{ f(x)}} \bigr) \,$.
• Compare the previous two ideas!
  To ‘graph the function $\,f\,$’ means to show all points of the form $\,\bigl(x,f(x)\bigr)\,$.
  To ‘graph the equation $\,y=f(x)\,$’ means to show all points of the form $\,\bigl(x,f(x)\bigr)\,$.
  These two requests mean exactly the same thing!

• Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
  Points on the graph of $\,y=3f(x)\,$ are of the form $\,\bigl(x,3f(x)\bigr)\,$.
  Thus, the graph of $\,y=3f(x)\,$ is found by taking the graph of $\,y=f(x)\,$, and multiplying the $\,y$-values by $\,3\,$.
  This moves the points farther from the $\,x$-axis, which makes the graph steeper.
• Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
  Points on the graph of $\,y=\frac13f(x)\,$ are of the form $\,\bigl(x,\frac13f(x)\bigr)\,$.
  Thus, the graph of $\,y=\frac13f(x)\,$ is found by taking the graph of $\,y=f(x)\,$, and multiplying the $\,y$-values by $\,\frac13\,$.
  This moves the points closer to the $\,x$-axis, which makes the graph flatter.
• Transformations involving $\,y\,$ work the way you would expect them to work—they are intuitive.
• Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$
  and asked about the graph of $\,y=3f(x)\,$: $$ \begin{align} \cssId{s57}{\text{original equation:}} &\quad \cssId{s58}{y=f(x)}\cr\cr \cssId{s59}{\text{new equation:}} &\quad \cssId{s60}{y=3f(x)} \end{align} $$ $$ \begin{gather} \cssId{s61}{\text{interpretation of new equation:}}\cr\cr \cssId{s62}{\overset{\text{the new y-values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s63}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}} \cssId{s64}{\overset{\quad\text{three times}\quad}{\overbrace{ \strut \ \ 3\ \ }}} \cssId{s65}{\overset{\qquad\text{the previous y-values}\quad}{\overbrace{ \strut\ \ f(x)\ \ }}} \end{gather} $$
• Summary of vertical scaling:
  
  Let $\,k \gt 1\,$.
  Start with the equation $\,y=f(x)\,$.
  Multiply the previous $\,y\,$-values by $\,k\,$, giving the new equation $\,y=kf(x)\,$.
  The $\,y$-values are being multiplied by a number greater than $\,1\,$, so they move farther from the $\,x$-axis.
  This makes the graph steeper, and is called a vertical stretch.
  
  Let $\,0 \lt k \lt 1\,$.
  Start with the equation $\,y=f(x)\,$.
  Multiply the previous $\,y\,$-values by $\,k\,$, giving the new equation $\,y=kf(x)\,$.
  The $\,y$-values are being multiplied by a number between $\,0\,$ and $\,1\,$, so they move closer to the $\,x$-axis.
  This makes the graph flatter, and is called a vertical shrink.
  
  In both cases, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,k\,b)\,$ on the graph of $\,y=kf(x)\,$.
  This transformation type is formally called vertical 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 counter-intuitive—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 $\,y=f(3x)\,$: $$ \begin{align} \cssId{sb16}{\text{original equation:}} &\quad \cssId{sb17}{y=f(x)}\cr\cr \cssId{sb18}{\text{new equation:}} &\quad \cssId{sb19}{y=f(3x)} \end{align} $$ $$ \begin{gather} \cssId{sb20}{\text{interpretation of new equation:}}\cr\cr \cssId{sb21}{y = f( \overset{\text{replace x by 3x}}{\overbrace{ \ \ 3x\ \ }}} ) \end{gather} $$ 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 $\,k\,x\,$ to give the new equation $\,y=f(k\,x)\,$. 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 $\,y=f(k\,x)\,$. 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 $\,(k\,a,b)\,$ on the graph of $\,y=f(\frac{x}{k})\,$. This transformation type is formally called horizontal scaling (stretching/shrinking).

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 $\ldots$

vertical scaling:
going from   $\,y=f(x)\,$   to   $\,y = kf(x)\,$   for   $\,k\gt 0$

horizontal scaling:
going from   $\,y = f(x)\,$   to   $\,y = f(k\,x)\,$   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.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Reflections and the Absolute Value Transformation