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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 $\,y=-f(x)\,$ are of the form $\,\bigl(x,-f(x)\bigr)\,$.
  Thus, the graph of $\,y=-f(x)\,$ is found by taking the graph of $\,y=f(x)\,$, and multiplying the $\,y$-values by $\,-1\,$.
  This reflects the graph about the $\,x$-axis.
• 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 $\,y=-f(x)\,$: $$ \begin{align} \cssId{s40}{\text{original equation:}} &\quad \cssId{s41}{y=f(x)}\cr\cr \cssId{s42}{\text{new equation:}} &\quad \cssId{s43}{y=-f(x)} \end{align} $$ $$ \begin{gather} \cssId{s44}{\text{interpretation of new equation:}}\cr\cr \cssId{s45}{\overset{\text{the new y-values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s46}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}} \cssId{s47}{\overset{\quad\text{-1 times}\quad}{\overbrace{ \strut \ \ -\ \ }}} \cssId{s48}{\overset{\qquad\text{the previous y-values}\quad}{\overbrace{ \strut\ \ f(x)\ \ }}} \end{gather} $$
• In reflection about the $\,x$-axis, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,-b)\,$ on the graph of $\,y=-f(x)\,$.

Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$. Points on the graph of $\,y=f(-x)\,$ are of the form $\,\bigl(x,f(-x)\bigr)\,$. How can we locate these desired points $\,\bigl(x,f(-x)\bigr)\,$? Pick a value of $\,x\,$. First, go to the point $\,\color{red}{\bigl(-x\,,\,f(-x)\bigr)}\,$ 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 $\,-x\,$, but the desired $\,x$-value is just $\,x\,$. Thus, the current $\,\color{purple}{x}$-value must be multiplied by $\,\color{purple}{-1}\,$; that is, each $\,\color{purple}{x}$-value must be sent to its opposite. The $\,\color{purple}{y}$-value remains the same. This causes the point to reflect about the $\,y$-axis, and gives the desired point $\,\color{green}{\bigl(x,f(-x)\bigr)}\,$. Thus, the graph of $\,y=f(-x)\,$ is the same as the graph of $\,y=f(x)\,$, except that it has been reflected about the $\,y$-axis.

Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$ and asked about the graph of $\,y=f(-x)\,$: $$ \begin{align} \cssId{s62}{\text{original equation:}} &\quad \cssId{s63}{y=f(x)}\cr\cr \cssId{s64}{\text{new equation:}} &\quad \cssId{s65}{y=f(-x)} \end{align} $$ $$ \begin{gather} \cssId{s66}{\text{interpretation of new equation:}}\cr\cr \cssId{s67}{y = f( \overset{\text{replace x by -x}}{\overbrace{ \ \ -x\ \ }}} ) \end{gather} $$ Replacing every $\,x\,$ by $\,-x\,$ in an equation causes the graph to be reflected about the $\,y$-axis. In reflection about the $\,y$-axis, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(-a,b)\,$ on the graph of $\,y=f(-x)\,$.

• Points on the graph of $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
  Points on the graph of $\,y=|f(x)|\,$ are of the form $\,\bigl(x,|f(x)|\bigr)\,$.
• Thus, the graph of $\,y=|f(x)|\,$ is found by taking the graph of $\,y=f(x)\,$ and taking the absolute value of the $\,y$-values.
  
  Points with positive $\,y$-values stay the same, since the absolute value of a positive number is itself.
  That is, points above the $\,x$-axis don't change.
  
  Points with $\,y=0\,$ stay the same, since the absolute value of zero is itself.
  That is, points on the $\,x$-axis don't change.
  
  Points with negative $\,y$-values will change, since taking the absolute value of a negative number makes it positive.
  That is, any point below the $\,x$-axis reflects about the $\,x$-axis.
  
  These actions are summarized by saying that ‘any part of the graph below the $\,x$-axis flips up’.
• Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$
  and asked about the graph of $\,y=|f(x)|\,$: $$ \begin{align} \cssId{s82}{\text{original equation:}} &\quad \cssId{s83}{y=f(x)}\cr\cr \cssId{s84}{\text{new equation:}} &\quad \cssId{s85}{y=|f(x)|} \end{align} $$ $$ \begin{gather} \cssId{s86}{\text{interpretation of new equation:}}\cr\cr \cssId{s87}{\overset{\text{the new y-values}}{\overbrace{ \strut\ \ y\ \ }}} \cssId{s88}{\overset{\text{are}}{\overbrace{ \strut\ \ =\ \ }}} \cssId{s89}{\overset{\quad\text{the absolute value of the previous y-values}\quad}{\overbrace{ \strut\ \ |f(x)|\ \ }}} \end{gather} $$
• In the absolute value transformation, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,|b|)\,$ on the graph of $\,y=|f(x)|\,$.

reflecting about the $\,x$-axis:
going from $\,y = f(x)\,$ to $\,y = -f(x)$

reflecting about the $\,y$-axis:
going from $\,y = f(x)\,$ to $\,y = f(-x)$

absolute value transformation:
going from $\,y = f(x)\,$ to $\,y = |f(x)|$
Any part of the graph on or above the $\,x$-axis stays the same;
any part of the graph below the $\,x$-axis flips up.

Make sure you see the difference between $\,y = -f(x)\,$ and $\,y = f(-x)\,$!

In the case of $\,y = -f(x)\,$, the minus sign is ‘on the outside’;
we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then multiplying by $\,-1\,$.
This is reflection about the $\,x$-axis.

In the case of $\,y = f(-x)\,$, the minus sign is ‘on the inside’;
we're multiplying $\,x\,$ by $\,-1\,$ before dropping it into the $\,f\,$ box.
This is reflection about the $\,y$-axis.

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

When you're done practicing, move on to:
Graphical Transformations: All Mixed Up