![vertical stretch vertical stretch](https://i.ytimg.com/vi/pXHcwo2zsQw/maxresdefault.jpg)
Students have to see the connections, and they have to see them over time.Īrithmetic aside, a second goal for all of my math classes centers on math as the science of patterns. Exploring again with logarithms reinforces the initial algebra, and creates strong, EXPLICIT parallels between exponential and logarithmic algebra–a point missed by many sources. 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. 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. But I make sure to keep spiraling back to the topic to reinforce the ideas. I spend a couple days on this algebra and then move on to other topics and applications. Exponential and log algebra is very useful in some scenarios, but I think teachers err when they drive to algebraic mastery in first encounters. How much time I spend in this algebra varies depending on the level of the class. 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.Įnjoy the strange transformational world of exponential and log functions! The sum property of logarithms proves the existence of this equally strange property:
![vertical stretch vertical stretch](https://thecollegepanda.com/sat-function-transformations-definitive-guide/stretched-vertically.png)
Here’s a Desmos page demonstrating the log property. But for logarithms, vertical stretches do morph the curve into a different shape. In this case, that means the transformation property that did not work for exponentials does work for logarithms! That is,Īny horizontal stretch of any logarithmic function is congruent to some vertical translation of the original function. 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.īut if a property is true for a function, then the inverse of the property generally should be true for the inverse of the function. I encourage them to play with the algebra or create another graph to investigate. My students inevitably ask if the same is true for horizontal stretches and vertical slides of exponentials. 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. 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.
![vertical stretch vertical stretch](https://i.ytimg.com/vi/5a0_f1xb-tQ/maxresdefault.jpg)
The implications of this are pretty deep. 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. Try any positive stretch you like, and you will always be able to find some horizontal translation that gives you the exact same result. That’s a pretty strange result if you think about it. By the time you change h to -2, the horizontal translation aligns perfectly with the vertical stretch. Now play with the h slider in line 6 to see if you can achieve the same results with a horizontal translation. Let’s say I wanted to quadruple the height of my function, so I move the a slider to 4. Likewise, the line 5 black graph is a horizontal translation of the parent, and the translation is controlled by the line 6 slider.
VERTICAL STRETCH FREE
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.įrom 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. I set up a Desmos page to explore this property dynamically (shown below).
![vertical stretch vertical stretch](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200813/CNX_Precalc_Figure_01_05_0242.jpg)
It is an inevitable result from the multiplication of common bases implies add exponents property: Not when exponentials are vertically stretched. Doesn’t stretching a curve by definition change its curvature? You can vertically stretch any exponential function as much as you want, and the shape of the curve will never change!īut that doesn’t make any sense.
VERTICAL STRETCH SOFTWARE
I use Desmos in this post, but this can be reproduced on any graphing software with sliders. 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. OK, this post’s title is only half true, but transforming exponentials can lead to counter-intuitive results.