The analytic view

Here is a video!

In analytic terms a rigid translation is when the relation that represents the function is “changed universally’ in the following way: for any two $x$ and $y$ with $f(x) = y$, whatever is being done to that pair of related points, the same thing is being done to every single pair of related points. Moreover, this change must always be a matter of adding or subtracting (in a particular way which we will discuss below), not multiplying or dividing. This is easier to explain starting with an example.

Notice how, in our example, we talked about the translations that applied to the $x$-value, and translations that applied to the $y$-value separately. This separation is made even more clear when we write the translation in functional notation. Let’s consider a new function $f(x)$ whose graph is;

We now want to apply the rigid translation that adds 4 to the $x$ and subtracts 5 from the $y$-value. We can write the result of this translation by first giving it a name, say $g$, and defining it as a modification of the original $f$. This turns out to be slightly more tricky than one may think however.

Intuitively we might try to write our translation as follows; $g(x) = f(x+4) - 5$. After all, we wanted to add $4$ to the $x$ and subtract $5$ from the $y$, so this seems pretty reasonable. Let’s see what the graph $f(x+4) - 5$ looks like.

A near miss! It appears that we got the vertical movement correct but it shifted the wrong direction horizontally; it shifted left instead of right! The specifics on why this horizontal shift is backwards are outside the scope of this course, but in this specific context it suffices to remember that anything that effects the $x$ is the opposite of what you’d expect. The mantra to remember is “Everything about $x$ is backward”. Since you would expect that we want to add 4 to $x$ (to move it to the right) what we actually want to do (in order to move it to the right) is subtract 4. Thus the $g(x)$ we actually want to use is $g(x) = f(x-4)-5$ whose graph is;

In summary:

Writing this rigid translation using function notation (ie $f(x-4)-5$) is the ‘functional representation’. In general, a rigid translation of a function $f$ can be written by adding (or subtracting) values from the functional argument (the part inside the parentheses that follow the letter $f$), or by adding/subtracting values to the overall result after applying $f$. So, if you want to add some number $c$ to the $x$ values and another number $k$ to the $y$ value, you can write the translation as;

$$f(x - c) + k$$

Notice that the $c$ inside has a subtraction sign in front of it. If you remember it in this form then you can write the $c$ as you would expect (since the $c$ is being subtracted not added which will take care of the ’opposite of what you’d except’ part). Thus if you want to move the graph to the right 4 units, you can use $c=4$ in the form, giving $f(x - c) = f(x - (4))$. In comparison if you wanted to move the graph to the left by 4 you would do the same process. Since it’s to the left you would think $c = -4$, so plugging that into the form above we have $g(x) = f(x - c) = f(x - (-4) ) = f(x+4)$ which is correct.

Play with the following graph to see what happens as you change the parameters. Be careful to observe the effect of the sign of the parameters in the example. Hopefully using this interactive graph will help make clear how changing the values (rigidly) translates the graph!