OK, this post’s title is only half true, but transforming exponentials can lead to counter-intuitive results. This post shares a cool transformations activity using dynamic graphing software–a perfect set-up for a mind-bending algebra or precalculus student lesson in the coming year. I use Desmos in this post, but this can be reproduced on any graphing software with sliders.
You can vertically stretch any exponential function as much as you want, and the shape of the curve will never change!
But that doesn’t make any sense. Doesn’t stretching a curve by definition change its curvature?
The answer is no. Not when exponentials are vertically stretched. It is an inevitable result from the multiplication of common bases implies add exponents property:
I set up a Desmos page to explore this property dynamically (shown below). The base of the exponential doesn’t matter; I pre-set the base of the parent function (line 1) to 2 (in line 2), but feel free to change it.
From its form, the line 3 orange graph is a vertical stretch of the parent function; you can vary the stretch factor with the line 4 slider. Likewise, the line 5 black graph is a horizontal translation of the parent, and the translation is controlled by the line 6 slider. That’s all you need!
Let’s say I wanted to quadruple the height of my function, so I move the a slider to 4. Now play with the h slider in line 6 to see if you can achieve the same results with a horizontal translation. By the time you change h to -2, the horizontal translation aligns perfectly with the vertical stretch. That’s a pretty strange result if you think about it.
Of course it has to be true because . Try any positive stretch you like, and you will always be able to find some horizontal translation that gives you the exact same result.
Likewise, you can horizontally slide any exponential function (growth or decay) as much as you like, and there is a single vertical stretch that will produce the same results.
The implications of this are pretty deep. Because the result of any horizontal translation of any function is a graph congruent to the initial function, AND because every vertical stretch is equivalent to a horizontal translation, then vertically stretching any exponential function produces a graph congruent to the unstretched parent curve. That is, any vertical stretch of any exponential will never change its curvature! Graphs make it easier to see and explore this, but it takes algebra to (hopefully) understand this cool exponential property.
NOT AN EXTENSION
My students inevitably ask if the same is true for horizontal stretches and vertical slides of exponentials. I encourage them to play with the algebra or create another graph to investigate. Eventually, they discover that horizontal stretches do bend exponentials (actually changing base, i.e., the growth rate), making it impossible for any translation of the parent to be congruent with the result.
ABSOLUTELY AN EXTENSION
But if a property is true for a function, then the inverse of the property generally should be true for the inverse of the function. In this case, that means the transformation property that did not work for exponentials does work for logarithms! That is,
Any horizontal stretch of any logarithmic function is congruent to some vertical translation of the original function. But for logarithms, vertical stretches do morph the curve into a different shape. Here’s a Desmos page demonstrating the log property.
The sum property of logarithms proves the existence of this equally strange property:
Hopefully the unexpected transformational congruences will spark some nice discussions, while the graphical/algebraic equivalences will reinforce the importance of understanding mathematics more than one way.
Enjoy the strange transformational world of exponential and log functions!