# 1.9 Graphing  (Page 2/3)

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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.

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

This violates the only restriction on functions—the rule of consistency . Remember that the $x$ -axis is the independent variable, the $y$ -axis the dependent. In this case, one “input” value $\left(3\right)$ is leading to two different “output” values $\left(-9,9\right)$ 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 $y$ -values for the same $x$ -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 $y=f\left(x\right)$ .

We can see from the graph that the domain of the graph is $-3\le x\le 6$ and the range is $-3\le y\le 2$ .

Question: What does the graph of $y=f\left(x\right)+2$ look like ?

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

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

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.

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 $y=\sqrt{x}$ 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.

who was the first nanotechnologist
k
Veysel
technologist's thinker father is Richard Feynman but the literature first user scientist Nario Tagunichi.
Veysel
Norio Taniguchi
puvananathan
Interesting
Andr
I need help
Richard
anyone have book of Abdel Salam Hamdy Makhlouf book in pdf Fundamentals of Nanoparticles: Classifications, Synthesis
what happen with The nano material on The deep space.?
It could change the whole space science.
puvananathan
the characteristics of nano materials can be studied by solving which equation?
sibaram
synthesis of nano materials by chemical reaction taking place in aqueous solvents under high temperature and pressure is call?
sibaram
hydrothermal synthesis
ISHFAQ
how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
STM - Scanning Tunneling Microscope.
puvananathan
how did you get the value of 2000N.What calculations are needed to arrive at it
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