What is Slope: Definition and 779 Discussions

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical.

A line is increasing if it goes up from left to right. The slope is positive, i.e.


{\displaystyle m>0}
A line is decreasing if it goes down from left to right. The slope is negative, i.e.


{\displaystyle m<0}
If a line is horizontal the slope is zero. This is a constant function.
If a line is vertical the slope is undefined (see below).The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
In mathematical language, the slope m of the line is











{\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}
The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function



{\displaystyle m=\tan(\theta )}
Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.
As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.
This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

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  1. Theexploer

    A sphere held steady on a slope by a rope

    The sum of the forces should be 0. Sin A'C'B' = px/b px = mg . sin alpha P should be px = - m.g. sin alpha and py = m.g.cos alpha Finally i fund as result F = -0.8 and R = -1.23 but for the second question i didn't fund the radius of the circle.
  2. L

    Slope of a mountain ridge (Gradient)

    Hi I am not quite sure if I have calculated the whole task correctly, since I am not sure whether I have solved task e correctly and unfortunately have problems with task f The function h(r) looks like this ##h(r)=\frac{x}{\sqrt{x^2+y^2}}+1## I got the following for the gradient ##\nabla...
  3. L

    Block on slope. Different answers using energy vs forces

    The first equation is when I use forces. The block is in static equilibrium, therefore the spring force should balance the gravitational force. The second equation is when I use energy principles. Energy before compression = Energy at compression. The height before is x * sintheta, and the...
  4. guyvsdcsniper

    Engineering Bode Plot Slope based on frequencies

    From 0 to ##10^3## ##\omega## there is a dB gain, from ##10^3## to ##10^5## there is another. Finally from ##10^5## to infinity the slope is constant (0). I know the formula $$dbV= 20log_{10}\frac{V_2}{V_1}$$ can give me the slope but that is in terms of Volts, but I have frequency and the...
  5. S

    Which material has the greatest Young's Modulus?

    My answer is (B) since it has the highest slope for the straight line part of the graph but the answer key is (A). Is it because the slope of graph B will decrease until the value less than slope A? So we don't only consider the straight line part but the whole graph? Thanks
  6. E

    B Calculating 'Typical' Grade of a Hiking Trail

    Hi all. I am tasked with quantifying hiking trail grades (percent grade) and their 'typical' grade. I am not sure if 'typical' is really a math term, but my first inclination was that it equates to 'average.' However, I now think there are two ways to approach this with different results. Is it...
  7. baby_1

    The relation between the normal and the slope of a cylindrical curve

    As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here: $$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$ I got confused because I don't have the curve equation(regarding the slope of the curve...
  8. C

    How to find coefficient of friction of a body sliding down the slope?

    So basically I need to find the coefficient of friction given the listed information. What bothers me is that I am getting two different accelerations for two different approaches. When I calculate acceleration using Fg=mgsin60 I do it this way: Fg=mgsin60 -> ma=mgsin60 ->a=gsin60 -> a=8.66. But...
  9. B

    Simple overestimate of slope uncertainty in regression

    Hi all, I am a science educator in high school. I have been thinking about how to make a simple estimate that 1st and maybe 2nd year students can follow for the propagation of error to the uncertainty of the slope in linear regression. The problem is typically that they make some measurements...
  10. C

    A Negative slope in Williamson-Hall model

    Good afternoon, I am analyzing a diffraction diagram (XRD) corresponding to a powder diffraction experiment , and I have obtained a negative slope value when plotting ##\left(\beta_{exp}-\beta_{inst}\right)\cos\theta## vs ##\sin\theta##. This implies that the strain coefficient is negative for...
  11. paulimerci

    Newton's Laws of motion -- Bicyclist pedaling up a slope

    F parallel - F applied - rolling resistance = ma I don't know how to calculate for rolling resistance. If the bicycle is not slipping rather it is rolling, should I ignore rolling resistance? And if I ignore that I would get, F parallel - F applied = ma F applied = F parallel -ma...
  12. hagopbul

    I Drawing Direction Fields for Higher Order ODEs

    Hello : Trying to find references on drawing direction fields of higher order differential equation by hand as 1st step then by computer , do you know any reference I can read ( PDF , books ,...) , and hope it is not only some short notes Best regards HB
  13. Rikudo

    A composite body on a slope

    Here is the hint that the book gave me: "For the maximum value of μ, the rod must be to the extreme right i.e. horizontally rightwards of the axis of the pipe" I think what it meant is the same as this: Note: in the calculation below, ##r## is the distance from the center of pipe to the CoM...
  14. Svelte1

    How is the slope of the shear force equal to the negative of the load?

    At any point between A and C the point load is negative (downwards), in the shear force diagram: positive is upwards, so this slope is negative. The equation says the slope should be positive. Is this something to do with shear force sign convention?
  15. James Brown

    B Will a round-headed rod topple if it slides down a frictionless slope?

    Well, the problem is that, someone told me that a ball won't roll when sliding down a frictionless slope because the resultant force mgsinx is parallel to the slope which means that the ball will slide down the slope. Now, replace the ball with a round headed rod, does this means that the rod...
  16. G

    Projectile motion when kicking a ball that lands on an upward slope

    We know x = R =max range (28m) on level ground. Need to find v()^2. Subbing y=0 into (1) above, get v(0)^2 = (gR^2/)/(2*cos (theta)^2 * tan (theta). ... (2) This didn't seem right, since this means v(0)^2 is a negative number ... maybe my orientation or algebra wrong? Anyway, didn't see...
  17. Eobardrush

    Ball rolling down a slope problem: Find an expression for time taken

    Question: Galileo released a metal ball from rest so that it could roll down a smooth inclined plane. The time t taken to roll a distance s was measured. He repeated the experiment, each time recording the time taken to travel a different fraction of the distance s. Write an expression for the...
  18. MrS

    I Open channel flow: derive function for energy slope on friction?

    I want to determine the normal flow depth in a perfectly horizontal circular conduit. The system characteristics are known (Internal pipe diameter, Mannings roughness, Discharge). However, I am not sure how to calculate the normal flow depth. When using Manning's equation one can find the normal...
  19. Physics_learner

    Kinematics - can't find the initial velocity according to the image

    I tried to write the data I understood from the image: y0=160m yf=0 x0=0 x1=192m I tried to express the total change in time using the position over time equation on the Y direction: y(t)=y0+v0y*(t2-t1)-0.5a(t2-t1)^2 but then I stuck with 2 variables and didn't know what to do any help?
  20. Hendra

    Ball rolling down a slope ending with a loop

    Hello everyone! I tried to solve this problem in a non-inertial system. Probably I should use the principle of conservation of mechanical energy in the following form $$mgH = \frac{3mgR}{2} + \frac{mV^2}{2}.$$ So the only thing to do is to compute $V^2$. I tried to find this value using the...
  21. chwala

    Find the time taken to travel from top of slope to base - Mechanics

    Now this is a textbook example with solution. I understand working to solution...my only reservation is on how they used acceleration. The cyclist, i understand was traveling at a constant acceleration of ##2## ##m/s^2## before reaching the top part of the slope. Now, if he is descending...
  22. speedythespeed

    B Oberth effect on a slope - experiment idea

    I'm just curious if something like the Oberth effect on a slope is doable as an experiment. I have a picture of my idea of what to do just looking for some opinions.
  23. Nick Tucker

    Calculating Force and Work for a Penguin Climbing up an Ice Slope

    Got a question from my science exam that I'm not sure how to figure it out. All the context I was given is attached. My attempt: Mass=26kg 26a = Force Work = 26a x 2 Work = 52aNot sure how to figure it out, as 52a is the wrong answer.
  24. mncyapntsi

    Finding tension at an angle

    Hi! I really can't figure this one out... I have a = (F-cos(36.1)g) = a and from that I get T = mB1 a = 6.3 (20.3736) = 128 N. Could someone please help? Thanks!
  25. N

    Find Slope of Secant Line

    Slope of a Tangent Line For f (x) = x^2 − 1: Find the slope of the secant line containing the points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)). Solution: I think I need to find f(-1). f(-1) = (-1)^2 - 1 f(-1) = 1 - 1 f(-1) = 0 Point P becomes (-1, 0). I now must find f(-1 + h)...
  26. N

    Find Slope of Secant Line

    Slope of a Tangent Line For f (x) = x^3. (a) Find the slope of the secant line containing the points (2, 8) and (3, 27). (b) Find the slope of the secant line containing the points (2, 8) and (x, f (x)), where x does not equal 2. For (a), I just have to find m, the slope using m = delta...
  27. F

    I Calculate m (slope) of a line given the angle in degrees

    Hi, I want calculate the m = slope of a linear line WHEN I already know the angle in degrees of the line. Here is an example: I calculate with Excel the angle with the following function: =+DEGREES(ATAN(0.0874887)) and I get as result the angle of 5%. But how do I calculate the value...
  28. The Bill

    History of "percent grade" for slope esp. in civil engineering?

    What is the origin and/or history of the usage of the term "percent grade" for 100 times slope in, especially but not exclusively, civil engineering? The combinations of search terms I've tried so far only bring back the definition, examples of how to use quantities in this format, and other...
  29. M

    MHB Word Problem for slope intercept graph equation

    I have this word problem that is asking for two different answers, the equation for the data and to calculate the shipping rate. I'm not understanding how to address either of the questions. Will someone please help me with this answer?
  30. M

    MHB Slope Intercept Equation with parallel

    I'm not understanding this question at all and am not sure how to even begin answering this. Any help would be appreciated. Write the slope-intercept equation of the line that is parallel to -9x-7y=4 and has the same y-intercept as the graph of -5x+11y=-22.
  31. Y

    Estimating head loss from pipe slope?

    I am trying to calculate head loss for a sloped pipe. I found this calculation here, which seems to be what I want..or at least a start: https://www.pumpsandsystems.com/pumps/april-2015-calculating-head-loss-pipeline My confusion/skepticism arises from the fact that the equations in the above...
  32. L

    Physics Graph Word Problem -- Motion of a person skiing down a slope

    I was able to do the first graph knowing that acceleration is 9.8 and my distance goes up by 10s (y-axis) and my time goes up by 1s (x-axis). For the other 3 graphs, I'm not sure where to begin because I don't know how to figure out my velocity
  33. karush

    MHB -apc.4.1.2 Find the slope at x=4

    $\tiny{apc.4.1.2}$ Find the slope of the tangent line to the graph of $f(x)=-x^2+4\sqrt{x}$ at $x=4$ $a.\ 8\quad b.\ -10\quad c.\ -9\quad d.\ -5\quad e.\ -7$ however not asked for here but I forgot how to find $b$ of $y=-7x+b$
  34. Jojo-11

    Why does acceleration increase by less as angle of slope increases?

    My only guess is that this is due to air resistance. Below an example of the predicted graph:
  35. karush

    MHB Finding Slope with Desmos and Table

    ok attemped to do this desmos but was sondering if there is away to get these slope in a 3rd column in the table with $m=\dfrac{\delta y}{\delta x}$
  36. mcastillo356

    A skier goes down a slope -- Ignoring friction, calculate the acceleration

    a) ##a_y=\dfrac{\sum{F_y}}{m}=\dfrac{N-mg\cos{\alpha}}{m}=(1-\cos{\alpha})g## ##a_x=\dfrac{\sum{F_x}}{m}=\dfrac{mg\sin{\alpha}}{m}=g\sin{\alpha}## ##a_y=(1-0,866)9,81\;m/s^2=1,31\;m/s^2## ##a_x=(0,5)9,81\;m/s^2=4,91\;m/s^2## How can it be a perpendicular acceleration?; which coordinate system am...
  37. P

    Automotive Vehicle Parking Brake Test on a Slope [How Convert to a Flat Ground test?]

    Vehicle Mass : 3 tonnes Manufacturer's instruction is to test vehicle's parking brake effectiveness is to engage parking brake on 30.96 degrees slope. If it holds, it is effective. I do not have 30.96 degrees slope & I want to replicate the same test on flat ground. I was thinking of...
  38. Zero

    A Spring pushing an object up a 25 cm slope

    f= kx = 150 n/m (.3m) = 45n Ug= 1/2 kx^2 = 1/2 (150) (.3)^2 = 75 (.09) = 6.75 j Ek= 1/2mv^2 6.75= 1/2 (.54) (v^2) 6.75= .27 (v^2) 25= v^2...
  39. U

    B How can I linearize my (negative slope) quadratic graph?

    I am doing an investigation of how much a beam sags, based on the distance from its midpoint. This is my hypothetical equation: The relationship between distance, d and sag, s is not a linear relationship. Below, is the determined relationship between the variables, linearized by natural...
  40. MathDestructor

    Mechanics: Slope Friction Problem

    So far, I have this: For Part ii) I know that: mgh = xmgsin(theta), but I don't know how to go further
  41. Z

    Subthreshold slope and transconductance

    I'm trying to understand the difference between subthreshold slope and transconductance. The subthreshold slope is the change in drain current / change in gate voltage. The transconductance of a MOSFET is proportional to the drain current / VGS - Vth. It would seem the subthreshold slope is...
  42. karush

    MHB 9.2.2 AP Calculus Exam -- slope field for which DE

    ok probably if one did a lot of these this could be solved by observation others separate the variables and take to Integral to get the equation
  43. karush

    MHB 9.2.2 AP Calculus Exam Slope Fields

    I'm just going to post this image now since my tablet won't render the latex. This is a free response question.. But my experience is that the methods of solving are more focused here at mhb saving many error prone steps.. Mahalo ahead...
  44. E

    How to convert a slope to an acceleration?

    Hello, I hope you are all very well. I am in second year of High School and I have a practical work in physics. The experiment was to release a long tape with a mass of 40g at the end from a certain height. An instrument would hit the tape 50 times/s and put a mark each time. From that we have...