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A teacher's guide to permutations and functions.

The end of the “Graphing” homework sets this topic up.

Have one person in the class “be” x 2 . He is allowed to use a calculator; so you can, for instance, hand him the number 1.7 and he will square it, producing the point (1.7,2.89).

Another person is x 2 + 1 . He is not allowed to use a calculator, but he is allowed to talk to the first person, who is. So if you hand him 1.7, he asks the first person, who says 2.9, and then he comes back with a 3.9. Make sure everyone understands what we have just learned: the graph of x 2 + 1 contains the point (1.7,3.9). Do a few points this way.

Another person is ( x + 1 ) 2 with the same rules. So if you give him a 1.7 he hands a 2.7 to the calculator person. Make sure everyone understands how this process gives us the point (1.7,7.3).

Talk about the fact that the first graph is a vertical permutation: it messed with the y-values that came out of the function. It’s easy to understand what it did. It added 1 to every y-value, so the function went up 1.

The second graph is a horizontal permutation: it messed with the x-values that went into the function. It’s harder to see what that did: why did ( x + 1 ) 2 move to the left? Ask them to explain that.

Now hand them the worksheet “Horizontal and Vertical Permutations I.” Hand one to each person—they will start in class, but probably finish in the homework. It’s on the long side.

The next day, talk it all through very carefully. Key points to bring out:

  1. What does f ( x ) + 2 mean? It means first plug a number into f ( x ) , and then add 2.
  2. And what does that do to the graph? It means every y-value is two higher than it used to be, so the graph moves up by 2.
  3. What does f ( x + 2 ) mean? It means first add 2, then plug a number into f ( x ) .
  4. And what does that do to the graph? It means that when x = 3 you have the same y-value that the old graph had when x = 5 . So your new graph is to the left of the old one.
  5. What does all that have to do with our rock? This should be a long-ish conversation by itself. The vertical and horizontal permutations represent very different types of changes in the life of our rock. Suggest a different scenario, such as our old standard, the number of candy bars in the room as a function of the number of students, c ( s ) . What scenario would c ( s ) + 3 represent? How about c ( s + 3 ) ?

If you haven’t already done so, introduce graphing on the calculator, including how to properly set the window. It only takes 5-10 minutes, but is necessary for the homework.

Now, put the graph of y = x 2 on the board. We saw what ( x + 1 ) 2 and x 2 + 1 looked like yesterday. What do you think x 2 would look like? How about ( x + 2 ) 2 3 ?

At some point, during the first or second day, you can come back to the idea of domain. What is the domain of y = x + 3 size 12{ sqrt {x+3} } {} ? See if they can see the answer both numerically (you can plug in x = 3 but not x = 4 ) and graphically (the graph of y = x size 12{ sqrt {x} } {} moved three spaces to the left, and its domain moved too).


“Horizontal and Vertical Permutations II”

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
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Commplementary angles
Idrissa Reply
im all ears I need to learn
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
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