A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. Such changes are a subset of the possible “transformations” of a function, and can be accomplished through vertical translation, horizontal translation, or both.

Imagine a graph that has been drawn on tracing paper that was loosely lain over a printed set of axes. If you move the tracing paper left, right, up, or down some distance, without rotating it in any way, you are “translating” the graph.

It is very useful to be able to interpret a function as a translation of a “parent function”. Understanding the behavior of

$g(x)=(x-3)^2+1$

is much easier if you think of it as the graph of a simpler looking function that you should already be very familiar with

$f(x)=x^2$

shifted right by 3, and up by 1.

This approach helps us understand the behavior ofmany functions, and makes complex-looking equations much easier to interpret.

Vertical Translation

Consider the graph of

$f(x)=2x$

The line that this describes passes through the origin and has a slope of two.

What happens to the graph of this line if the equation is changed by adding a three to it?

$g(x)=2x+3$

Note that this new function can be described in terms of the first:

$g(x)=f(x)+3$

which shows that the y-coordinate of every point on the graph of g(x) is exactly 3 higher than the corresponding point on f(x). Adding the “+3″ to the original “rule for finding f(x)” caused the entire function to be “translated vertically” by a positive three.

This process works for any function:

$g(x)=x^2+3\\*~\\*f(x)=x^2~~is~the~parent~function,~so\\*~\\*g(x)=f(x)+3$

thus $g(x)$ is $f(x)$ translated vertically by +3.

Another example:

$g(x)=sin(x)-10\\*~\\*f(x)=sin(x)~~is~the~parent~function,~so\\*~\\*g(x)=f(x)-10$

thus $g(x)$ is $f(x)$ translated vertically by -10.

Another example:

$g(x)=\frac{1}{x}+k\\*~\\*f(x)=\frac{1}{x}~~is~the~parent~function,~so\\*~\\*g(x)=f(x)+k$

thus $g(x)$ is $f(x)$ translated vertically by k

Horizontal Translation

Consider the same function described at the beginning of the Vertical Translation section:

$f(x)=2x$

This equation describes a line that passes through the origin, and has a slope of two.

What happens to the graph if the equation is changed by replacing every “x” in the equation by (x-4):

$g(x)=f(x-4)\\*~\\*g(x)=2(x-4)$

Consider one point on the graph of $f(x):(1,2)$. Since we are examining horizontal translations, let’s figure out what value of “x” will cause $g(x)$ to have the same result (y-coordinate) that $f(x)$ had at x=1:

$f(1)=2\\*~\\*g(x)=2\\*~\\*2(x-4)=2\\*~\\*(x-4)=1\\*~\\*x=5$

So, $g(5)=f(1)=2$. The point (1,2) on the graph of $f(x)$, along with all other points on the graph, has been shifted horizontally by +4.

But why has it been shifted right (by +4) when an “x-4″ was substituted for each “x”? Our eyes are easily drawn to the “- 4″, and it is very tempting to assume that it describes the direction of the translation just as it did with vertical translations above.

The statement

$g(x)=f(x-4)$

from above relates the two functions. It tells us that $g(x)$ will have the same result as $f(x)$ when the input to $f(x)$ is 4 less than (to the left of) the input to $g(x)$.  If $f(x)$ is to the left of $g(x)$, then $g(x)$ must have been moved to the right of the $f(x)$ “parent function”.

If you prefer a procedural approach to finding the direction of a horizontal translation, solve the transformation expression for zero:

$(x-4)=0\\*~\\*x=+4$

and the result will always give your the magnitude and direction of the translation (see Keep Your Eye On The Variable).

This process works for any function:

$g(x)=(x+3)^2\\*~\\*f(x)=x^2~~is~the~parent~function,~so\\*~\\*g(x)=f(x+3)$

thus $g(x)$ is a horizontal translation of $f(x)$ by -3.

Another example:

$g(x)=sin(x-10)\\*~\\*f(x)=sin(x)~~is~the~parent~function,~so\\*~\\*g(x)=f(x-10)$

thus $g(x)$ is a horizontal translation of $f(x)$ by +10.

Another example:

$g(x)=(x+5)^2-3(x+5)+7\\*~\\*f(x)=x^2-3x+7~~is~the~parent~function,~so\\*~\\*g(x)=f(x+5)$

thus $g(x)$ is a horizontal translation of $f(x)$ by -5. Note that every instance of “x” in $f(x)$ must be changed to (x+5) for $g(x)$ to represent a horizontal translation of $f(x)$ by -5. The notation $g(x)=f(x+5)$ describes this idea very compactly and elegantly, but don’t forget to substitute (x+5) in for every instance of “x” in $f(x)$.

One last example:

$g(x)=\frac{1}{x+k}\\*~\\*f(x)=\frac{1}{x}\\*~\\*g(x)=f(x+k)$

thus $g(x)$ is $f(x)$ translated horizontally by -k.

Reconciling Horizontal And Vertical Translations

If all of the above has made sense so far, you may have noticed $f(x)+3$ translated the function in a positive vertical direction, yet $f(x+3)$ translates the function in a negative horizontal direction. This difference in sign was a source of many errors in my work until I realized I could reconcile this difference by treating both independent and dependent variables in the same manner.

Consider

$g(x)=(x+5)^2-3(x+5)+7$

If I subtract the 7 from both sides, it becomes:

$(g(x)-7)=(x+5)^2-3(x+5)$

Since every instance of $g(x)$ occurs as a $g(x)-7$, and every instance of “x” occurs as $(x+5)$, you may treat “x” and $g(x)$ as having both been translated relative to a parent function, and you may analyze them both in exactly the same manner:
– what value of $g(x)$ makes $(g(x)-7)=0$? Positive 7. So the translation in the $g(x)$ direction is positive 7.
– what value of “x” makes $(x+5)=0$? Negative 5. So there is a translation in the “x” direction of negative 5.

Therefore, the $g(x)$ function shown above describes two translations of a “parent function” $f(x)$:

$f(x)=x^2-3x\\*~\\*(g(x)-7)=f(x+5)$

One translation is by -5 horizontally (in the “x” direction) and the other by +7 vertically (in the $g(x)$ direction).

Equivalent Translations

In mathematics, it is often (but not always) possible to produce the same end result in different ways. When working with linear equations, the same translation can be achieved as follows:

$f(x)=x\\*~\\*g(x)=f(x)+3=(x)+3~~a~vertical~translation~of~+3\\*~\\*h(x)=f(x+3)=(x+3)~~a~horizontal~translation~of~-3$

Note that both the horizontal and vertical translations above produced the same final equation when simplified: $x+3$. If you imagine the graph of f(x), which will be a line with a slope of one that passes through the origin, then move the line up 3, note that this translation will also shift the x-intercept left to -3.

Similarly, if you start with the graph of f(x) again, but shift the graph left by 3, note that the y-intercept moves up to +3. This shows that, in the case of lines, the same end result (a vertical translation by 3 in this case) can be achieved by either a horizontal translation, a vertical translation, or a combination of one translation in each direction: left 2 and up 1, left 4 and down 1, right 2 and up 5, etc.

If the slope of the line were not one, things would get a bit more interesting, as different translation magnitudes would be required – depending on the direction:

$f(x)=2x\\*~\\*g(x)=f(x)+4=(2x)+4=2x+4~~a~vertical~translation~of~+4\\*~\\*h(x)=f(x+2)=2(x+2)=2x+4~~a~horizontal~translation~of~-2$

In this example, where the line has a slope of 2, a vertical translation must have the opposite sign and twice the magnitude of a horizontal translation to have the same effect. This makes sense once you also know how to analyze function dilations.