Shifting function graphs: writing vertical offset on the y-side?

Question

Students tend to mix up signs when shifting function graphs around: consider $y=x^2$. To shift it one unit upwards ("increasing $y$"), you write $y=x^2+1$, to shift it to the right ("increasing $x$"), you write $y=(x-1)^2$. On the one hand, it's $+$, on the other hand you have to use $-$ for somewhat "the same thing".

It recently came to my mind that the interpretation of the function graph as all points fulfilling the relation $ \mathbb R^2\supset R = \{(x, y) ~|~ y=x^2\}$ might be more helpful for this matter:

• To shift the graph vertically, you use $R_\text{v} = \{(x, y) ~|~ y+a=x^2\}$.
• To shift horizontally: $R_\text{h} = \{(x, y) ~|~ y=(x+a)^2\}$
• Negative values for $a$ shift in increasing direction for both cases.

Is there any literature or experience on this approach?

Answer 1

Imagine the simple equation of a straight line: $ y=2x$.
You want to move this up one point (in the $Y$-direction), and you get: $y=2x+1$.

If you want to move this up one point in the $X$-direction, then you first need to write this with $x$ at one side: $x=y/2$.
So you get : $x=y/2+1$.
Writing this back to the $y=f(x)$ notation becomes: $y=2(x-1)$.

So, the appearance of the minus is a simple consequence of the fact that we write our functions in the notation $y=f(x)$.

Answer 2

Sorry, I don’t know of any literature but I trialed this approach with a class recently. I felt strongly that applying different approaches to transformations in the x and y directions was less than helpful when they would subsequently meet, for example, circles where the centre is other than the origin and which can be considered as transformations. It didn’t work for the majority; the idea of adding a constant to the right hand side to increment y by that amount was too intuitive for them to relinquish.

My next approach will be to keep the changes to the right side of the equals sign with translations of a in the x direction resulting in $f(x-a)$ and translations of b in the y direction resulting in $f(x)+b$. I will go beck to making the link and showing consistency in the two directions when I later introduce circles.

Answer 3

The only "literature" I can think of is how nearly every algebra textbook I've seen presents the "point-slope" form of a linear function: $$y - y_1 = m(x - x_1)$$

Here, the translation of axes is applied the same to both variables.

Answer 4

According to my experience concrete examples like the following are best suited to support the understanding. Starting from the axes $y = 0$ and $x = 0$ I have always shown the following, one after another of course.