# 2.5 Function concepts -- lines

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This module discusses lines and their uses, and slope.

Most students entering Algebra II are already familiar with the basic mechanics of graphing lines. Recapping very briefly: the equation for a line is $y=\text{mx}+b$ where $b$ is the $y$ -intercept (the place where the line crosses the $y$ -axis) and m is the slope. If a linear equation is given in another form (for instance, $4x+2y=5$ ), the easiest way to graph it is to rewrite it in $y=\text{mx}+b$ form (in this case, $y=-2x+2\frac{1}{2}$ ).

There are two purposes of reintroducing this material in Algebra II. The first is to frame the discussion as linear functions modeling behavior . The second is to deepen your understanding of the important concept of slope.

Consider the following examples. Sam is a salesman—he earns a commission for each sale. Alice is a technical support representative—she earns $100 each day. The chart below shows their bank accounts over the week. After this many days (t) Sam’s bank account (S) Alice’s bank account (A) 0 (*what they started with)$75 $750 1$275 $850 2$375 $950 3$450 $1,050 4$480 $1,150 5$530 $1,250 Sam has some extremely good days (such as the first day, when he made$200) and some extremely bad days (such as the second day, when he made nothing). Alice makes exactly $100 every day. Let d be the number of days, S be the number of dollars Sam has made, and A be the number of dollars Alice has made. Both S and A are functions of time. But $s\left(t\right)$ is not a linear function , and $A\left(t\right)$ is a linear function . Linear Function A function is said to be “linear” if every time the independent variable increases by 1, the dependent variable increases or decreases by the same amount . Once you know that Alice’s bank account function is linear, there are only two things you need to know before you can predict her bank account on any given day. • How much money she started with ($750 in this example). This is called the $y$ - intercept .
• How much she makes each day ($100 in this example). This is called the slope . $y$ -intercept is relatively easy to understand. Verbally, it is where the function starts; graphically, it is where the line crosses the $y$ -axis. But what about slope? One of the best ways to understand the idea of slope is to convince yourself that all of the following definitions of slope are actually the same. Definitions of Slope In our example In general On a graph Each day, Alice’s bank account increases by 100. So the slope is 100. Each time the independent variable increases by 1, the dependent variable increases by the slope. Each time you move to the right by 1, the graph goes up by the slope. Between days 2 and 5, Alice earns$300 in 3 days. 300/3=100.Between days 1 and 3, she earns \$200 in 2 days. 200/2=100. Take any two points. The change in the dependent variable, divided by the change in the independent variable, is the slope. Take any two points. The change in $y$ divided by the change in $x$ is the slope. This is often written as $\frac{\mathrm{\Delta y}}{\mathrm{\Delta x}}$ , or as $\frac{\text{rise}}{\text{run}}$
The higher the slope, the faster Alice is making moey. The higher the slope, the faster the dependent variable increases. The higher the slope, the faster the graph rises as you move to the right.

So slope does not tell you where a graph is, but how quickly it is rising. Looking at a graph, you can get an approximate feeling for its slope without any numbers. Examples are given below.

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
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
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?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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