



Key concepts
 Linear functions may be graphed by plotting points or by using the
y intercept and slope. See
[link] and
[link] .
 Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See
[link] .
 The
y intercept and slope of a line may be used to write the equation of a line.
 The
x intercept is the point at which the graph of a linear function crosses the
x axis. See
[link] and
[link] .
 Horizontal lines are written in the form,
$f(x)=b.$ See
[link] .
 Vertical lines are written in the form,
$x=b.$ See
[link] .
 Parallel lines have the same slope.
 Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See
[link] .
 A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the
x  and
y values of the given point into the equation,
$f(x)=mx+b,$ and using the
$b$ that results. Similarly, the pointslope form of an equation can also be used. See
[link]
.
 A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See
[link] and
[link] .
 A system of linear equations may be solved setting the two equations equal to one another and solving for
$x.$ The
y value may be found by evaluating either one of the original equations using this
x value.
 A system of linear equations may also be solved by finding the point of intersection on a graph. See
[link] and
[link] .
Section exercises
Verbal
If the graphs of two linear functions are parallel, describe the relationship between the slopes and the
y intercepts.
The slopes are equal;
y intercepts are not equal.
If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the
y intercepts.
If a horizontal line has the equation
$f\left(x\right)=a$ and a vertical line has the equation
$x=a,$ what is the point of intersection? Explain why what you found is the point of intersection.
The point of intersection is
$\left(a,a\right).$ This is because for the horizontal line, all of the
$y$ coordinates are
$a$ and for the vertical line, all of the
$x$ coordinates are
$a.$ The point of intersection will have these two characteristics.
Explain how to find a line parallel to a linear function that passes through a given point.
Explain how to find a line perpendicular to a linear function that passes through a given point.
First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation
$y=mx+b$ and solve for
$b.$ Then write the equation of the line in the form
$y=mx+b$ by substituting in
$m$ and
$b.$
Algebraic
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
$\begin{array}{l}4x7y=10\hfill \\ 7x+4y=1\hfill \end{array}$
Questions & Answers
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?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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?
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
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
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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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