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In this section, you will:
  • Graph functions using vertical and horizontal shifts.
  • Graph functions using reflections about the x -axis and the y -axis.
  • Determine whether a function is even, odd, or neither from its graph.
  • Graph functions using compressions and stretches.
  • Combine transformations.
Figure_01_05_038
(credit: "Misko"/Flickr)

We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.

Graphing functions using vertical and horizontal shifts

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

Identifying vertical shifts

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift , moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function g ( x ) = f ( x ) + k , the function f ( x ) is shifted vertically k units. See [link] for an example.

Figure_01_05_001
Vertical shift by k = 1 of the cube root function f ( x ) = x 3 .

To help you visualize the concept of a vertical shift, consider that y = f ( x ) . Therefore, f ( x ) + k is equivalent to y + k . Every unit of y is replaced by y + k , so the y - value increases or decreases depending on the value of k . The result is a shift upward or downward.

Vertical shift

Given a function f ( x ) , a new function g ( x ) = f ( x ) + k , where k is a constant, is a vertical shift    of the function f ( x ) . All the output values change by k units. If k is positive, the graph will shift up. If k is negative, the graph will shift down.

Adding a constant to a function

To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. [link] shows the area of open vents V (in square feet) throughout the day in hours after midnight, t . During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

Figure_01_05_002

We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in [link] .

Figure_01_05_003a

Notice that in [link] , for each input value, the output value has increased by 20, so if we call the new function S ( t ) , we could write

S ( t ) = V ( t ) + 20

This notation tells us that, for any value of t , S ( t ) can be found by evaluating the function V at the same input and then adding 20 to the result. This defines S as a transformation of the function V , in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See [link] .

t 0 8 10 17 19 24
V ( t ) 0 0 220 220 0 0
S ( t ) 20 20 240 240 20 20

Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
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Maciej
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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.
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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
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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
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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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
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Yasmin
what is the function of carbon nanotubes?
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
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preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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good afternoon madam
AMJAD
what is system testing
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what is the application of nanotechnology?
Stotaw
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
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anybody can imagine what will be happen after 100 years from now in nano tech world
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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
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
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silver nanoparticles could handle the job?
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Azam
Hello
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I'm interested in Nanotube
Uday
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Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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