# Learning about finding the slope of a line

I just don't get it!

We're learning about finding the slope of a line. And I don't get any of it, so can someone explain it to me? I could find the slope through a pair of points:

C->1(2,1) 1-1 =0
D->2(3,1) 2-3 =-1 and the answer is 0...(note that this is algebra((or pre-algebra)), not calculus)

but when it comes to the graph, I'm not so sure. My guess would be to do the samething I did above? (there are multiple xy coordinates on the graphs in my textbook).

Thanks!

Zefram
I'm sorry, what's your question? What a slope of 0 means graphically? Or the deal with the whole concept of slope in general?

I meant to ask how to measure the slope on a graph.[?]

Mulder
To find the slope of a straight line through two coordinates find the change in the y-coordinate and divide by the change in x-coordinate. You'll probably need to be more specific...

[sigh], I guess I'll try to ask my teacher
(and he's not the best at explaining) ...i can't be specific on something I don't understand...

Is my work(on my very first post on this thread) similar to what you're specifying?

***A spark of hope***

Mulder

Actually I don't follow this notation...
Originally posted by MajinVegeta
C->1(2,1) 1-1 =0
Can you explain what you have tried to do?

Zefram
Slope on a graph is rise over run. So if you've got a line that moves up three units every time you move over one unit in the positive x direction, then the slope will be the rise (3) over the run (1) = 3/1 = 3. That's a steeper slope than, say, a line that moves up one unit (rise) every time it moves over three units (run) in the positive x direction--this slope would be 1/3.

Ugh, that was pretty nonmathematical sounding, huh?

I GOT IT! now for questions:

is the run always 1?

Actually I don't follow this notation...

quote:
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Originally posted by MajinVegeta
C->1(2,1) 1-1 =0e

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Can you explain what you have tried to do?

actually, the two notations are one. there was supposed to be
a fraction bar seperating the notations.

Staff Emeritus
Gold Member
No the run does not have to be 1, it is just easy to work with. If the slope is something like 3/5, that means up 3 when your run is 5, you could say the run is 2.5, and rise is 1.5. the key is that the ratio remains the same for a given line. That means a line has a constant slope. No matter where you measure it you will get the same result. No matter how big your run the RATIO of rise/run remains the same.

can the slope be angular(consist of right angles in particular)?

Staff Emeritus
Gold Member
Yes, to get the angle for rise over run, simply take arctan (rise/run)

tangent is opposite side over adjacent side to the angle you are measuring. The opposite side is the rise, the adjacent side is the run.

The right angles depend.

If the angle is 0, that would correspond to a slope of 0.

If the angle is 90, then the slope is undefined, because it is no longer a function (multiple values in y for a specific value x)

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We're learning about finding the slope of a line. And I don't get any of it, so can someone explain it to me?
The "slope" of a line basically tells you how steep the line is.

I tried drawing some lines as examples but the HTML yanks out spaces so it didn't work - sorry.

The sign of a lines slopw indicates whether the line slopes up or down [ where up and down means starting at the left most part of the line and seeing if the point on the line moves upr or down as you move to the right]. A "+" sign means it slopes up and a "-" sign means it slopes down. If it has no slope (i.e. slope = 0) then it netier goes up nor down. Thus the line

***********

has zero slope. To measure the slope we look at a particular portion of the line - Then we use the "change in rise" and compare it to the "change in run" for that part of the line. And as you can see just by looking at a line it doesn't matter what part of the line or how much of the line you choose. To make this clear use the symbols

dy = "Change in rise"
dx = "Change in run"

and define then as follows: For *any* two points A and B on the line A(X_a, Y_b) and B(X_b, Y_b) where A is a point to the left of B. By "to the left of B" I mean that we choose A as the one such that X_a < X_b. If X_a = X_b then the line is said to have infinite slope (infinite steepness).

dx = X_b - X_a
dy = Y_b - Y_a

Then "define" the quantity "m" as the slope and it has the value

m = dy/dx

Play with it a bit and try using different points. If you have further questions I'd be more than happy to explain further.

Pete

m = dy/dx

what is that equation for? I'm guessing is to measure an infinite slope?

thanks a lot you guys!

now, for transversal slopes...I'm not sure what they are..I know what a transversal line is but not a transversal slope (are they the same?)

arctan (rise/run)

so "arctan" means rise/run?

OKAY,functions! I'm not 2 sure about them either (I don't have the best math teacher...)

Staff Emeritus
Gold Member
As to m=dy/dx...

Originally posted by MajinVegeta
what is that equation for? I'm guessing is to measure an infinite slope?

No, that's for any slope. I'm guessing that pmb doesn't know how to make a "delta" in this forum (neither do I). It simply says that the slope (m) of a line is equal to the change in y over the change in x.

so "arctan" means rise/run?

No, "slope" means rise/run. "arctan(x)" is the inverse of tan(x). I seriously doubt that you need to worry about it.

Mentor
what is that equation for? I'm guessing is to measure an infinite slope?

dy = delta y = change in y = rise
dx = delta x = change in x = run
m = slope

dx/dy = rise/run = slope = m

That notation is generally not used until calculus though. And people here tend to have trouble lowering the level of their explanations (no offense, but it is a real problem and not just in here).

The simplest way I can give you for slope between two points:

(x1,y1),(x2,y2)

(y2-y1)/(x2-x1) = slope

On a graph, you can pick any points along the line, but its best to pick points that make the math as easy as possible. If the line starts at the origin (0,0) use that point for your first point.

No, "slope" means rise/run. "arctan(x)" is the inverse of tan(x). I seriously doubt that you need to worry about it.

Why shouldn't I worry about it?
what's tan(x)?

Zefram
Originally posted by MajinVegeta
Why shouldn't I worry about it?

You're in pre-algebra, correct? You won't need to know that for a while.

Well, I still want to learn about. I'm 13, into theo. physics...and I'm not supposed to worry about whether or not the universe is infinite or if time travel using closed time loops is possible or not unitl I'm MUCH, MUCH older. Preferrably, the only way I am going to learn easily would be to go at MY speed, which is very rapid. If no one here won't tell me, I will learn what it is sooner or later (it will be soon!)

Staff Emeritus
Gold Member
Originally posted by MajinVegeta
Well, I still want to learn about.

And you will, in good time.

In the first post of your thread you said, in reference to slopes of lines, "I just don't get it!"

That means you've got a way to go before learning about trigonometry, which is prerequisite for the inverse trigonometric functions, about which you are now inquiring.

Your curiosity is great, but there is no illuminating way to render a 'pop' account of trigonometry, as there is in theoretical physics. Learning math and physics is like building a house: It all starts with a solid foundation.

Master algebra and geometry. Then you will get to trig.

That's my $0.02. Okay, Tom, you're right! I've moderately mastered the concept today. Okay, it's no longer a weak point. So, what else should I start learning next? Suggestions? Staff Emeritus Science Advisor Gold Member This thread reminds me of a fact I've experienced over the course of my (still relatively short) life: The more I learn, the more I realize I don't yet know. - Warren Staff Emeritus Science Advisor Gold Member Originally posted by MajinVegeta So, what else should I start learning next? Suggestions? Yes, finish the book. You say that your preferred pace is 'rapid', right? OK, so outpace the class if you need to. Do as many problems as you can. Ask questions. Then, just like a for...next loop in a computer program, do that for as many math and physics courses as you can. The normal progression in upper-track high school math is like this: *Algebra *Plane and Solid Geometry *College Algebra/Trig (also called Precalculus) *Calculus with Analytic Geometry *Multivariable Calculus (if you're advanced enough. I wasn't ) Something I did to help myself out was get into Schaum's Outlines, which are the greatest thing since sliced bread. They are an inexpensive (~$15/each) "Cliff's Notes" version of a textbook, but the fact that they have a ton of solved problems makes up for the lack of detail. In fact, many details are reserved for the solved problems section. I learned algebra-based physics this way, before even taking the course.

If your coursework is not enough for you, then I say get the one entitled Geometry. It has no prerequisites except a logical mind.

There, that's my $0.04. STAii Majin. If you want to know about arctan(), i will be happy to tell you. But first you have to tell us, do you know what a function is ? Do you know the meaning of inverse functions ? And do you know trigonometry ? Staff Emeritus Science Advisor Gold Member Originally posted by Zefram You're in pre-algebra, correct? You won't need to know that for a while. DOH. My bad! I shouldn't have brought it up. When you were asking about slopes, I thought you were in pre-cal (and thus would have had at least some trig already) My memory of my early high-school years must be getting fuzzy. I don't remember learning slopes before then. Originally posted by chroot This thread reminds me of a fact I've experienced over the course of my (still relatively short) life: The more I learn, the more I realize I don't yet know. - Warren Truth behold! I recall hearing that from the philosophy forum on PFs 2.0! It is true, for the more you learn, the more there is to learn. And it so happens that, for onece, such a miracle happens to one of my favirote things:learning. Originally posted by Tom Yes, finish the book. You say that your preferred pace is 'rapid', right? OK, so outpace the class if you need to. Do as many problems as you can. Ask questions. I'll ask here. Not my math teacher! He is good with math, but explaining is not his specialty. I'm glad I found pfs! Then, just like a for...next loop in a computer program, do that for as many math and physics courses as you can. What's a next loop computer program? The normal progression in upper-track high school math is like this: *Algebra *Plane and Solid Geometry *College Algebra/Trig (also called Precalculus) *Calculus with Analytic Geometry *Multivariable Calculus (if you're advanced enough. I wasn't ) Really? Tom, you're like really intelligent, now. I bet you're a black belt at multivariable calculus! Something I did to help myself out was get into Schaum's Outlines, which are the greatest thing since sliced bread. They are an inexpensive (~$15/each) "Cliff's Notes" version of a textbook, but the fact that they have a ton of solved problems makes up for the lack of detail. In fact, many details are reserved for the solved problems section. I learned algebra-based physics this way, before even taking the course.

I'll check'em out right away, sir!

If your coursework is not enough for you, then I say get the one entitled Geometry. It has no prerequisites except a logical mind.
[/b[

Well, I'll have to do that during the summer. My science and core(homeroom) teachers just LOVE piling us with project after project. So, that leaves me 2 hours a day, to go on pfs. aha! I'll do the math while I'm here! good idea, Tom!
[/QUOTE]

Originally posted by STAii
Majin.
If you want to know about arctan(), i will be happy to tell you.
But first you have to tell us, do you know what a function is ?
Do you know the meaning of inverse functions ?
And do you know trigonometry ?

no to all the above. I'd love to learn though!

Note, if an equation you bring up is related to physics, then I'll get it faster. Really weird huh?

*In the following summary, "_" should be read as "sub".

To summarize:

The Slope of a Straight Line

Say you have two points P_1 and P_2. Each point has its own coordinates: P_1(x_1, y_1); P_2(x_2, y_2) The difference between the points is (Delta x, Delta y) = (x_2 - x_1, y_2 - y_1). If Delta x is not 0, the slope of the line connecting the two points P_1 and P_2 is Delta y / Delta x.

Next topic: Functions

A function is a set of actions, performed on some input, that gives some output. The notation for the output of a function is F(T), where F is the name of the function and T is a list of inputs. (The generic function is f(x); you see f(x) a lot.) E.g., s(t) may be used to denote, say, the position of a train a time t.

arctan(rise / run) is a function: arctan is the name of the function and rise / run is the input. arctan can be used to convert a slope to an angle, i.e., a = arctan m, where m is the slope of a straight line and a is the angle that the line makes with the x-axis.

All of the following are functions:

f(x) = mx + b
f(x) = x^2
f(x) = e^x

Brief Calc Demo:

We can find the slope at any point on a function by using the straight line formula and shrinking the run to zero! We call this new function df(x)/dx

f(x) = x^2

df(x)/dx = rise / run, run —> 0

= (f(x + dx) - f(x)) / (dx)

= ((x + dx)^2 - (x)^2) / dx

= (x^2 + 2xdx + dx^2 - x^2) / dx

= (2xdx + dx^2) / dx

= 2x + dx

, and, since dx = run —> 0,

df(x)/dx = 2x

Originally posted by MajinVegeta
Okay, Tom, you're right! I've moderately mastered the concept today. Okay, it's no longer a weak point.
So, what else should I start learning next? Suggestions?

Are you sure you've completely mastered the concept? What if I were to ask you: a line has a slope of 1/4 and goes through the point (2,3); does this line go through the origin, and what is the x value of the line when it's y value equals 6?

If the method to get to the solutions to these questions isn't immediately obvious to you, you probably haven't yet mastered slope, and should probably try to master it a little more before moving on to something else. Like Tom was saying, math builds on itself, it's important that you have a good foundation to build on.

Slope Related Physics Questions for Magin

Originally posted by MajinVegeta
Note, if an equation you bring up is related to physics, then I'll get it faster. Really weird huh?

Okay Magin, here's a physics related question that oughta test to see how well you know your slopes (and don't forget to answer the "and why" part of the questions, that is the most important part in making sure you really understand the concept of slope!):

GIVEN: The velocity of some moving particle is found to be 2 meters per second and is constant over the period of time over which it is being observed.

QUESTION 1: If you were to make a graph of the position of the particle versus time what would the slope of the graph be and why?

QUESTION 2: If you were to make a graph of the acceleration of the particle versus time what would the graph look like and what would it's slope be and why?

What I meant was that if you were to teach me math in the context of physics, I would better understand it.

If you have a line that has points on it but it is too hard to calculate rise over run because the results you have are obscure (for instance the line of best fit in an experiment), then this is a simple formula that I found on http://richardbowles.tripod.com: [Broken]

[sum]X.[sum]Y - N.[sum]XY over ([sum]X)2 - N.[sum]X2

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