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Lesson 2: Slope

# 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!
• 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
• 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).
• 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?
• 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
• 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.