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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!).

## Want to join the conversation?

• 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?
• Increasing: the graph goes up from left to right

Decreasing: the graph goes down from left to right

Horizontal: the graph is perfectly flat (Δy = 0)

Vertical: the graph is perfectly straight up-and-down (Δx = 0)

Hope this helps!
• Ok y'all, I think I figured out how to easily see if a slope is negative or positive.
All you have to do is look to see which lines it passes through. If the line passes through both negative or both positive lines, it is negative. If the line passes through a positive and a negative line, then it is positive.
Though that was a cool little trick I'd share with you.
• There is a faster way... All lines that slant downward as they move left to right have negative slopes. All lines that slant upward as they move left to right have positive slope.

You didn't say how you would recognize if the slope wat positive or negative if it crosses at the origin (0,0). The more generalized approach I just gave covers this scenario and all others (except horizontal and vertical lines).
• 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)
• 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
• Does the slope line only have to be in the NE direction? Or can it be in the opposite direction, like NW?
• 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!
• How do you know when the slope is negative or positive?
• If the graph of a line rises from left to right, the slope is positive. If the graph of the line falls from left to right the slope is negative.
• Did you purposefully make lines 1, 2, and 4 (pink, blue, and orange) converge on the same point?
• Does infinite slope exist? What happens if the slope of the graph of the line that has a number that does not exist?
• Well technically, yes. Slope is Rise/Run. If the Rise is infinite and run is 1 that would make a line that is almost vertical. This is a very hypothetical situation though; I really have no idea what the practical application of this is.
• 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?
• Line might go up, doesn't change or go down
Up - Positive slope
No change - Undefined (you can't divide by 0)
Down - Negative slope
• I have one question. WHY
• I have one question for you ... Why not?
Look around you. There are many items in real life that involve slanted lines. All require an understanding of slope.
-- Is inflation going up or down?

Those are just a couple of quick examples. Do an internet search for where is slope used in real life.
• can you get a slope with a decimal point in it
• Yes. Say it costs \$4.65 per bag of chips at a store. You can form an equation, y=4.65x to tell you costs for x amount of chips.
You have to realize that you will see these decimals will be turned fractions more often than not because slope is a ratio of change in y/change in x. For money problems as above and some others, they are called unit rate ( \$4.65/1 bag) which are the ones most likely to be decimals.

## 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.