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But what about the shape of the graph? The graph shows a gradual incline up to 18", and then a precipitous drop back down to 12"; and this pattern repeats throughout the shown time. The most likely explanation is that Alice’s hair grows slowly until it reaches 18", at which point she goes to the hair stylist and has it cut down, within a very short time (an hour or so), to 12". Then the gradual growth begins again.

The rule of consistency, graphically

Consider the following graph.

A horizontal parabola opening up to the right where x = y-squared.

This is our earlier “U” shaped graph ( y = x 2 size 12{y=x rSup { size 8{2} } } {} ) turned on its side. This might seem like a small change. But ask this question: what is y size 12{y} {} when x = 3 size 12{x=3} {} ? This question has two answers. This graph contains the points ( 3, 9 ) size 12{ \( 3, - 9 \) } {} and ( 3,9 ) size 12{ \( 3,9 \) } {} . So when x = 3 size 12{x=3} {} , y size 12{y} {} is both 9 and –9 on this graph.

This violates the only restriction on functions—the rule of consistency . Remember that the x size 12{x} {} -axis is the independent variable, the y size 12{y} {} -axis the dependent. In this case, one “input” value ( 3 ) size 12{ \( 3 \) } {} is leading to two different “output” values ( 9,9 ) size 12{ \( - 9,9 \) } {} We can therefore conclude that this graph does not represent a function at all. No function, no matter how simple or complicated, could produce this graph.

This idea leads us to the “vertical line test,” the graphical analog of the rule of consistency.

The Vertical Line Test
If you can draw any vertical line that touches a graph in two places, then that graph violates the rule of consistency and therefore does not represent any function.

It is important to understand that the vertical line test is not a new rule! It is the graphical version of the rule of consistency. If any vertical line touches a graph in two places, then the graph has two different y size 12{y} {} -values for the same x size 12{y} {} -value, and this is the only thing that functions are not allowed to do.

What happens to the graph, when you add 2 to a function?

Suppose the following is the graph of the function y = f ( x ) size 12{y=f \( x \) } {} .

The sum of tow graphs. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {} ; Contains the following points (among others): ( 3,2 ) size 12{ \( - 3,2 \) } {} , ( 1, 3 ) size 12{ \( - 1, - 3 \) } {} , ( 1,2 ) size 12{ \( 1,2 \) } {} , ( 6,0 ) size 12{ \( 6,0 \) } {}

We can see from the graph that the domain of the graph is 3 x 6 size 12{ - 3<= x<= 6} {} and the range is 3 y 2 size 12{ - 3<= y<= 2} {} .

Question: What does the graph of y = f ( x ) + 2 size 12{y=f \( x \) +2} {} look like ?

This might seem an impossible question, since we do not even know what the function f ( x ) size 12{f \( x \) } {} is. But we don’t need to know that in order to plot a few points.

x size 12{x} {} f ( x ) size 12{f \( x \) } {} f ( x + 2 ) size 12{f \( x+2 \) } {} so y = f ( x ) size 12{y=f \( x \) } {} contains this point and y = f ( x ) + 2 size 12{y=f \( x \) +2} {} contains this point
–3 2 4 ( 3,2 ) size 12{ \( - 3,2 \) } {} ( 3,4 ) size 12{ \( - 3,4 \) } {}
–1 –3 –1 ( 1, 3 ) size 12{ \( - 1, - 3 \) } {} ( 1, 1 ) size 12{ \( - 1, - 1 \) } {}
1 2 4 ( 1,2 ) size 12{ \( 1,2 \) } {} ( 1,4 ) size 12{ \( 1,4 \) } {}
6 0 2 ( 6,0 ) size 12{ \( 6,0 \) } {} ( 6,2 ) size 12{ \( 6,2 \) } {}

If you plot these points on a graph, the pattern should become clear. Each point on the graph is moving up by two . This comes as no surprise: since you added 2 to each y-value, and adding 2 to a y-value moves any point up by 2. So the new graph will look identical to the old, only moved up by 2.

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}
The same graph as above shifted two places in the positive-y direction.
y = f ( x ) + 2 size 12{y=f \( x \) +2} {} ; All y size 12{y} {} -values are 2 higher

In a similar way, it should be obvious that if you subtract 10 from a function, the graph moves down by 10. Note that, in either case, the domain of the function is the same, but the range has changed.

These permutations work for any function . Hence, given the graph of the function y = x size 12{y= sqrt {x} } {} below (which you could generate by plotting points), you can produce the other two graphs without plotting points, simply by moving the first graph up and down.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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