1.9 Graphing  (Page 3/3)

 Page 3 / 3

Other vertical permutations

Adding or subtracting a constant from $f\left(x\right)$ , as described above, is one example of a vertical permutation: it moves the graph up and down. There are other examples of vertical permutations.

For instance, what does doubling a function do to a graph? Let’s return to our original function:

What does the graph $y=2f\left(x\right)$ look like? We can make a table similar to the one we made before.

$x$ $f\left(x\right)$ $2f\left(x\right)$ so $y=2f\left(x\right)$ contains this point
–3 2 4 $\left(-3,4\right)$
–1 –3 –6 $\left(-1,-6\right)$
1 2 4 $\left(1,4\right)$
6 0 0 $\left(6,0\right)$

In general, the high points move higher; the low points move lower. The entire graph is vertically stretched , with each point moving farther away from the x-axis.

Similarly, $y=\frac{1}{2}f\left(x\right)$ yields a graph that is vertically compressed, with each point moving toward the x-axis.

Finally, what does $y=-f\left(x\right)$ look like? All the positive values become negative, and the negative values become positive. So, point by point, the entire graph flips over the x-axis.

What happens to the graph, when you add 2 to the x value?

Vertical permutations affect the y-value; that is, the output, or the function itself. Horizontal permutations affect the x-value; that is, the numbers that come in. They often do the opposite of what it naturally seems they should.

Let’s return to our original function $y=f\left(x\right)$ .

Suppose you were asked to graph $y=f\left(x+2\right)$ . Note that this is not the same as $f\left(x\right)+2$ ! The latter is an instruction to run the function, and then add 2 to all results. But $y=f\left(x+2\right)$ is an instruction to add 2 to every x-value before plugging it into the function.

• $f\left(x\right)+2$ changes $y$ , and therefore shifts the graph vertically
• $f\left(x+2\right)$ changes $x$ , and therefore shifts the graph horizontally.

But which way? In analogy to the vertical permutations, you might expect that adding two would shift the graph to the right. But let’s make a table of values again.

$x$ $x+2$ $f\left(x+2\right)$ so $y=f\left(x+2\right)$ contains this point
–5 –3 f(–3)=2 $\left(-5,2\right)$
–3 –1 f(–1)=–3 $\left(-3,-3\right)$
–1 1 f(1)=2 $\left(-1,2\right)$
4 6 f(6)=0 $\left(4,0\right)$

This is a very subtle, very important point—please follow it closely and carefully! First of all, make sure you understand where all the numbers in that table came from. Then look what happened to the original graph.

The original graph $f\left(x\right)$ contains the point $\left(6,0\right)$ ; therefore, $f\left(x+2\right)$ contains the point $\left(4,0\right)$ . The point has moved two spaces to the left.

You see what I mean when I say horizontal permutations “often do the opposite of what it naturally seems they should”? Adding two moves the graph to the left .

Why does it work that way? Here is my favorite way of thinking about it. $f\left(x-2\right)$ is an instruction that says to each point, “look two spaces to your left, and copy what the original function is doing there .” At $x=5$ it does what $f\left(x\right)$ does at $x=3$ . At $x=\text{10}$ , it copies $f\left(8\right)$ . And so on. Because it is always copying $f\left(x\right)$ to its left , this graph ends up being a copy of $f\left(x\right)$ moved to the right . If you understand this way of looking at it, all the rest of the horizontal permutations will make sense.

Of course, as you might expect, subtraction has the opposite effect: $f\left(x-6\right)$ takes the original graph and moves it 6 units to the right . In either case, these horizontal permutations affect the domain of the original function, but not its range .

Other horizontal permutations

Recall that $y=2f\left(x\right)$ vertically stretches a graph; $y=\frac{1}{2}f\left(x\right)$ vertically compresses . Just as with addition and subtraction, we will find that the horizontal equivalents work backward.

$x$ $2x$ $f\left(2x\right)$ so $y=2f\left(x\right)$ contains this point
–1½ –3 2 $\left(-1\frac{1}{2},2\right)$
–½ –1 –3 $\left(-\frac{1}{2};-3\right)$
½ 1 2 $\left(\frac{1}{2};2\right)$
3 6 0 $\left(3,0\right)$

The original graph $f\left(x\right)$ contains the point $\left(6,0\right)$ ; therefore, $f\left(2x\right)$ contains the point $\left(3,0\right)$ . Similarly, $\left(-1;-3\right)$ becomes $\left(-\frac{1}{2};-3\right)$ . Each point is closer to the y-axis; the graph has horizontally compressed .

We can explain this the same way we explained $f\left(x-2\right)$ . In this case, $f\left(2x\right)$ is an instruction that says to each point, “Look outward, at the x-value that is double yours, and copy what the original function is doing there .” At $x=5$ it does what $f\left(x\right)$ does at $x=\text{10}$ . At $x=-3$ , it copies $f\left(-6\right)$ . And so on. Because it is always copying f(x) outside itself, this graph ends up being a copy of $f\left(x\right)$ moved inward ; ie a compression. Similarly, $f\left(\frac{1}{2}x\right)$ causes each point to look inward toward the y-axis, so it winds up being a horizontally stretched version of the original.

Finally, $y=f\left(-x\right)$ does precisely what you would expect: it flips the graph around the y-axis. $f\left(-2\right)$ is the old $f\left(2\right)$ and vice-versa.

All of these permutations do not need to be memorized: only the general principles need to be understood. But once they are properly understood, even a complex graph such as $y=-2\left(x+3{\right)}^{2}+5$ can be easily graphed. You take the (known) graph of $y={x}^{2}$ , flip it over the x-axis (because of the negative sign), stretch it vertically (the 2), move it to the left by 3, and move it up 5.

With a good understanding of permutations, and a very simple list of known graphs, it becomes possible to graph a wide variety of important functions. To complete our look at permutations, let’s return to the graph of $y=\sqrt{x}$ in a variety of flavors.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!