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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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