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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

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. Jun 17, 2014 Download for free at http://cnx.org/content/col11666/1.1
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