Graphing Tools: Vertical and Horizontal Translations (Part 2)

(This page is Part 2. Click here for Part 1.)

Ideas Regarding Horizontal Translations (Moving Left/Right)

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

  Points on the graph of $\,y=f(x+3)\,$ 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 $\,\color{red}{\bigl(x+3\,,\,f(x+3)\bigr)}\,$ 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.

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

  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 $\,y=f(x+3)\,$:

interpretation of new equation:

• 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 $\,y=f(x)\,.$ 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 $\,y=f(x+p)\,.$

Additionally: Start with the equation $\,y=f(x)\,.$ Replace every $\,x\,$ by $\,x-p\,$ to give the new equation $\,y=f(x-p)\,.$ 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 $\,y\,$ (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 Translations:
Going from $\,y=f(x)\,$ to $\,y = f(x) \pm c\,$

Horizontal Translations:
Going from $\,y = f(x)\,$ to $\,y = f(x\pm c)\,$

Make sure you see the difference between (say) $\,y = f(x) + 3\,$ and $\,y = f(x+3)\,$!

In the case of $\,y = f(x) + 3\,,$ 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 $\,y = f(x + 3)\,,$ 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