GRAPHING TOOLS:
VERTICAL AND HORIZONTAL TRANSLATIONS

There are things that you can DO to an equation of the form [beautiful math coming... please be patient] $\,y=f(x)\,$
that will change the graph in a variety of ways.

For example, you can move the graph up or down, left or right,
reflect about the [beautiful math coming... please be patient] $\,x\,$ or $\,y\,$ axes, stretch or shrink vertically or horizontally.

An understanding of these transformations makes it easy to graph a wide variety of functions,
by starting with a ‘basic model’ and then applying a sequence of transformations to change it to the desired function.

In this discussion, we will explore moving a graph up/down (vertical translations)
and moving a graph left/right (horizontal translations).

When you finish studying this lesson, you should be able to do a problem like this:

GRAPH: [beautiful math coming... please be patient] $\,y=(x-3)^2+5\,$

Here are ideas that are needed to understand graphical transformations.

IDEAS REGARDING FUNCTIONS AND THE GRAPH OF A FUNCTION

IDEAS REGARDING VERTICAL TRANSLATIONS (MOVING UP/DOWN)
IDEAS REGARDING HORIZONTAL TRANSLATIONS (MOVING LEFT/RIGHT)
  • Points on the graph of [beautiful math coming... please be patient] $\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
    Points on the graph of $\,y=f(x+3)\,$ are of the form $\,\bigl(x,f(x+3)\bigr)\,$.
  • How can we locate these desired points $\,\bigl(x,f(x+3)\bigr)\,$?

    First, go to the point $\,\bigl(x+3\,,\,f(x+3)\bigr)\,$ on the graph of $\,y=f(x)\,$.
    This point has the $\,y$-value that we want, but it has the wrong $\,x$-value.

    Move this point $\,3\,$ units to the left.
    Thus, the $\,y$-value stays the same, but the $\,x$-value is decreased by $\,3\,$.
    This gives the desired point $\,\bigl(x,f(x+3)\bigr)\,$.

    Thus, the graph of $\,y=f(x+3)\,$ is the same as the graph of $\,y=f(x)\,$, shifted LEFT three units.

    Thus, replacing $\,x\,$ by $\,x+3\,$ moved the graph LEFT (not right, as might have been expected!)
  • 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 $\,y=f(x+3)\,$: [beautiful math coming... please be patient] $$ \begin{align} \text{original equation:} &\quad y=f(x)\cr\cr \text{new equation:} &\quad y=f(x+3) \end{align} $$ [beautiful math coming... please be patient] $$ \begin{gather} \text{interpretation of new equation:}\cr\cr y = f( \overset{\text{replace x by x+3}}{\overbrace{ x+3}} ) \end{gather} $$ Replacing every $\,x\,$ by $\,x+3\,$ in an equation moves the graph $\,3\,$ units TO THE LEFT.
  • Summary of horizontal translations:
    Let $\,p\,$ be a positive number.

    Start with the equation [beautiful math coming... please be patient] $\,y=f(x)\,$.
    Replace every $\,x\,$ by $\,x+p\,$ to give the new equation $\,y=f(x+p)\,$.
    This shifts the graph LEFT $\,p\,$ units.
    A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a-p,b)\,$ on the graph of [beautiful math coming... please be patient] $\,y=f(x+p)\,$.

    Additionally:
    Start with the equation [beautiful math coming... please be patient] $\,y=f(x)\,$.
    Replace every $\,x\,$ by $\,x-p\,$ to give the new equation [beautiful math coming... please be patient] $\,y=f(x-p)\,$.
    This shifts the graph RIGHT $\,p\,$ units.
    A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a+p,b)\,$ on the graph of $\,y=f(x-p)\,$.

    This transformation type (shifting left and right) is formally called horizontal translation.
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 [beautiful math coming... please be patient] $\,y\,$
(that is, transformations that change the $\,y$-values of the points), we say:

DO THIS to the previous $\,y$-value.

For transformations involving [beautiful math coming... please be patient] $\,x\,$
(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 translations:
going from [beautiful math coming... please be patient] $\,y=f(x)\,$ to $\,y = f(x) \pm c\,$

horizontal translations:
going from [beautiful math coming... please be patient] $\,y = f(x)\,$ to $\,y = f(x\pm c)\,$

Make sure you see the difference between (say) [beautiful math coming... please be patient] $\,y = f(x) + 3\,$ and $\,y = f(x+3)\,$!

In the case of [beautiful math coming... please be patient] $\,y = f(x) + 3\,$, the $\,3\,$ is ‘on the outside’;
we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then adding $\,3\,$ to it.
This is a vertical translation.

In the case of [beautiful math coming... please be patient] $\,y = f(x + 3)\,$, the $\,3\,$ is ‘on the inside’;
we're adding $\,3\,$ to $\,x\,$ before dropping it into the $\,f\,$ box.
This is a horizontal translation.

EXAMPLES:
Question:
Start with $\,y = f(x)\,$.
Move the graph TO THE RIGHT $\,2\,$.
What is the new equation?
Solution:
This is a transformation involving $\,x\,$; it is counter-intuitive.
You must replace every $\,x\,$ by $\,x-2\,$.
The new equation is:
$\,y = f(x-2)\,$
Question:
Start with $\,y = x^2\,$.
Move the graph DOWN $\,3\,$.
What is the new equation?
Solution:
This is a transformation involving $\,y\,$; it is intuitive.
You must subtract $\,3\,$ from the previous $\,y\,$-value.
The new equation is:
$\,y = x^2 - 3\,$
Question:
Let $\,(a,b)\,$ be a point on the graph of $\,y = f(x)\,$.
Then, what point is on the graph of $\,y = f(x+5)\,$?
Solution:
This is a transformation involving $\,x\,$; it is counter-intuitive.
Replacing every $\,x\,$ by $\,x+5\,$ in an equation causes the graph to shift $\,5\,$ units to the LEFT.
Thus, the new point is $\,(a-5,b)\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Vertical and Horizontal Scaling
(stretching and shrinking)


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

Note: There are lots of questions like this:
“Start with $\,y = f(x)\,$.   Move UP $\,2\,$.   What is the new equation?”
Here is the same question, stated more precisely:
“Start with the graph of $\,y = f(x)\,$.   Move this graph UP $\,2\,$.   What is the equation of the new graph?”
All the ‘graphs’ are implicit in the problem statements.
(When something is implicit then it's understood to be there, even though you can't see it.)
(MAX is 64; there are 64 different problem types.)