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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 = mx + b size 12{y= ital "mx"+b} {} where b size 12{b} {} is the y size 12{y} {} -intercept (the place where the line crosses the y size 12{y} {} -axis) and m is the slope. If a linear equation is given in another form (for instance, 4x + 2y = 5 size 12{4x+2y=5} {} ), the easiest way to graph it is to rewrite it in y = mx + b size 12{y= ital "mx"+b} {} form (in this case, y = 2x + 2 1 2 size 12{y= - 2x+2 { { size 8{1} } over { size 8{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 ( t ) size 12{s \( t \) } {} is not a linear function , and A ( t ) size 12{A \( t \) } {} 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 size 12{y} {} - intercept .
  • How much she makes each day ($100 in this example). This is called the slope .

y size 12{y} {} -intercept is relatively easy to understand. Verbally, it is where the function starts; graphically, it is where the line crosses the y size 12{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 size 12{y} {} divided by the change in x size 12{x} {} is the slope. This is often written as Δy Δx size 12{ { {Δy} over {Δx} } } {} , or as rise run size 12{ { { ital "rise"} over { ital "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.

A Line with a positive slope of 1
A slope of 1: each time you go over 1, you also go up 1
A Line with a sttep positive slope of about 3 or 4
A steep slope of perhaps 3 or 4
A Line with a gentle positive slope of about 1/2
A gentle slope of perhaps 1 2 .
A horizontal line with a no slope
A horizontal line has a slope of 0: each time you go over 1, you don’t go up at all!
A Line with a steep negative slope of about -2
This goes down as you move left to right. So the slope is negative. It is steep: maybe a –2.

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Precalculus with engineering applications. OpenStax CNX. Jan 24, 2011 Download for free at http://cnx.org/content/col11267/1.3
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