# 1.10 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.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
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
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
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
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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