Exponentials Don’t Stretch

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.

THE SCENARIO

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:

b^a * b^c = b^{a+c}

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.

exp1

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.

exp2

Of course it has to be true because y = 2^{x-(-2)} = 2^x*2^2 = 4*2^x.  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.

exp3

The sum property of logarithms proves the existence of this equally strange property:

log(A) + log(x) = log(A*x)

CONCLUSION

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!

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3 responses to “Exponentials Don’t Stretch

  1. How much time do you spend with your students deeply in the algebra of exponential and logarithmic equations? It would seem that the point of introducing the fact that scaling in Y and translating in X are similar, would be to then master the algebra of exponential and logarithmic expressions, which are very common in many applications. I am curious as to how you tackle that part.

    • How much time I spend in this algebra varies depending on the level of the class. Exponential and log algebra is very useful in some scenarios, but I think teachers err when they drive to algebraic mastery in first encounters. I spend a couple days on this algebra and then move on to other topics and applications. But I make sure to keep spiraling back to the topic to reinforce the ideas. This is strongly supported by brain research which says that learning with time to forget and then re-exposing actually has a stronger influence on long-term recall. When I return with this particular lab 1-2 weeks after initial exposure to exponential algebra, I reinforce the algebra AND the sense of play and pattern discovery in math. Exploring again with logarithms reinforces the initial algebra, and creates strong, EXPLICIT parallels between exponential and logarithmic algebra–a point missed by many sources. Students have to see the connections, and they have to see them over time.

      Arithmetic aside, a second goal for all of my math classes centers on math as the science of patterns. When you notice something unusual happening, it is powerful to be able to explain WHY. For this scenario, almost every student has been disturbed by the fact that vertical stretches have no ability to shape-deform an exponential. This runs counter to everything they’ve learned about transformations: scale changes (not dilations) alter curves.

  2. Pingback: Applications of Logarithm Algebra | CAS Musings

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