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Positive & negative slope

Sal analyzes what it means for a slope to be positive or negative (spoiler: it affects the direction of the line!).

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  • winston baby style avatar for user Someone's Name
    If let's say you get a slope and it tells you that you need to describe the slope using words like Increasing, Decreasing, Horizontal and Vertical. how would you be able to define those words?
    (22 votes)
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  • sneak peak yellow style avatar for user jackxxu
    Why don't you do the slope as ∆x/∆y? Isn't that the same as the coordinates of a coordinate plane? Why do we have to do the slope as ∆y/∆x? I am very confused!!
    (1 vote)
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    • cacteye blue style avatar for user Jerry Nilsson
      It's because we describe 𝑦 as a function of 𝑥:
      𝑦 = 𝑚𝑥 + 𝑏

      If we have two points (𝑥₁, 𝑦₁) and (𝑥₂, 𝑦₂) we get the two equations
      𝑦₁ = 𝑚𝑥₁ + 𝑏
      𝑦₂ = 𝑚𝑥₂ + 𝑏

      Thereby,
      𝛥𝑦 = 𝑦₂ − 𝑦₁ =
      = 𝑚𝑥₂ + 𝑏 − (𝑚𝑥₁ + 𝑏) =
      = 𝑚𝑥₂ − 𝑚𝑥₁ = 𝑚(𝑥₂ − 𝑥₁) =
      = 𝑚 ∙ 𝛥𝑥 ⇒

      ⇒ 𝑚 = 𝛥𝑦∕𝛥𝑥
      (28 votes)
  • blobby green style avatar for user wiemejuru
    What if you don't have a whole graph and you just have one box how do you figure out if it's negative or positive?
    (6 votes)
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  • leafers ultimate style avatar for user EHCash
    Did you purposefully make lines 1, 2, and 4 (pink, blue, and orange) converge on the same point?
    (7 votes)
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  • aqualine sapling style avatar for user Sanjanaa V
    Does the slope line only have to be in the NE direction? Or can it be in the opposite direction, like NW?
    (2 votes)
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    • hopper cool style avatar for user Seed Something
      Great Question!

      No linear equation slope runs towards Northwest…
      but Negatives run from the Northwest to the Southeast, (downward to the right).

      ±Slopes of a linear equation can be measured in either direction, but the direction the line runs is from Left to Right.

      So either towards the Northeast or the Southeast.

      Positive slopes have an increasing slope that runs from lower left positions to upper right coordinates.
      (always kinda Northeast -ish).
      ↗️ Positive Slope
      is an 'increasing slope' because as x inputs become larger, the y outputs become larger too.

      Negative slopes have a decreasing slope, so they run from upper left positions towards lower right coordinates.
      (always kinda Southeast -ish).
      ↘️ Negative Slope
      is a 'decreasing slope' because as x inputs become larger, the y outputs become smaller.

      Both ↗️↘️ Positive and Negative sloped lines include all x and all y values. So every single number is on their lines!

      There's also:
      Zero Slope ↔️ a Horizontal Line, that includes all x-values, but only one y-value. As x increases or decreases y just stays the same. (So all possible x inputs map to the same y output.)
      Undefined Slope ↕️ a Vertical Line with only one x-value, to all y-values. Vertical line is the only one that doesn't work within a function, since an input must be unique to an output, but one x maps to all y).

      ★So with Linear Equations, it's just those four slope line types to learn and understand.

      Most of the time it will be about…
      ↗️Positive = increasing y outputs.
      ↘️Negative = decreasing y outputs.

      (≧▽≦) I hope that helps!
      (11 votes)
  • blobby green style avatar for user Katya Dudich
    I don't understand when he said the line was a slope of two when it wasn't touching any twos on the graph, pls help I am so lost ;-;
    (4 votes)
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    • mr pink green style avatar for user David Severin
      There are several issues, the first is that the domain of any linear function is all real numbers, so there has to be a value of y when x=2. However, slope does not have to do anything with a single point, but changes from one point to another point on the line. A slope of 2 means to get from one point to the next, you go up 2 (rise 2) and right 1 (run 1).
      (4 votes)
  • aqualine ultimate style avatar for user Himalaya Navarro
    Would a completely horizontal line have a slope of 0? If so, what would be the slope of a completely vertical line? Would it be undefined as you cannot divide by 0 (Δx = 0, Δy/Δx = undefined), or would it be infinity?
    (2 votes)
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  • blobby green style avatar for user gamikeska
    which slope is greater, m=4 or m=-5
    (3 votes)
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  • blobby green style avatar for user Bonngard Jayden
    why do we use slope man to tell witch slope is positive and witch is negative. is it based on witch way he runs?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      slope person is a graphic organizer to help remember, and as will all graphic organizers and mnemonic devices, we use them until we no longer need them any more because we know it. Positive slopes are based on as x increases, y is also increasing, so change in y/change in x is positive. Negative slopes are based on as x increases, y is decreasing, so we end up with a negative change in y/positive change in x which gives a negative answer.
      (3 votes)
  • piceratops sapling style avatar for user Pratyush soni
    can there be a slope value = e
    (2 votes)
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Video transcript

- [Voiceover] Slope is defined as your change in the vertical direction, and I could use the Greek letter delta, this little triangle here is the Greek letter delta, it means change in. Change in the vertical direction divided by change in the horizontal direction. That is the standard definition of slope and it's a reasonable way for measuring how steep something is. So for example, if we're looking at the xy plane here, our change in the vertical direction is gonna be a change in the y variable divided by change in horizontal direction, is gonna be a change in the x variable. So let's see why that is a good definition for slope. Well I could draw something with a slope of one. A slope of one might look something like... so a slope of one, as x increases by one, y increases by one, so a slope of one... is going to look like this. Notice, however much my change in x is, so for example here, my change in x is positive two, I'm gonna have the same change in y. My change in y is going to be plus two. So my change in y divided by change in x is two divided by two is one. So for this line I have slope is equal to one. But what would a slope of two look like? Well, a slope of two should be steeper and we can draw that. Let me start at a different point, so if I start over here a slope of two would look like... for every one that I increase in the x direction I'm gonna increase two in the y direction, so it's going to look like... that. This line right over here, you see it. If my change in x is equal to one, my change in y is two. So change in y over change in x is gonna be two over one, the slope here is two. And now, hopefully, you're appreciating why this definition of slope is a good one. The higher the slope, the steeper it is, the faster it increases, the faster we increase in the vertical direction as we increase in the horizontal direction. Now what would a negative slope be? So let's just think about what a line with a negative slope would mean. A negative slope would mean, well we could take an example. If we have our change in y over change in x was equal to a negative one. That means that if we have a change in x of one, then in order to get negative one here, that means that our change in y would have to be equal to negative one. So a line with a negative one slope would look like... would look like this. Notice, as x increases by a certain amount, so our delta x here is one, y decreases by that same amount instead of increasing. So now this is what we consider a downward sloping line. So change in y is equal to negative one. So our change in y over our change in x is equal to negative one over one which is equal to negative one. So the slope of this line is negative one. Now if you had a slope with negative two, it would decrease even faster. So a line with a slope of negative two could look something like this. So as x increases by one, y would decrease by two. So it would look something like... it would look like that. Notice, as our x increases by a certain amount, our y decreases by twice as much. So this right over here has a slope of negative two. So hopefully this gives you a little bit more intuition for what slope represents and how the number that we use to represent slope, how you can use that to visualize how steep a line is. A very high positive slope, as x increases, y is going to increase fairly dramatically. If you have a negative slope... as x increases, your y is actually going to decrease. And then the higher the slope, the steeper, the more you increase as x increases, and the more negative the slope, the more you decrease as x increases.