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Lesson 4: 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!!
• 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!
• 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
• Did you purposefully make lines 1, 2, and 4 (pink, blue, and orange) converge on the same point?
• 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 ;-;
• 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).
• 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?
• Yes, to both parts.
A horizontal line has a slope of 0.
A vertical line has a slope of undefined because the change in X is 0 and we can't divide by 0.
• which slope is greater, m=4 or m=-5
• m=4 is a greater slope, because it is a positive number. A positive number is always greater than a negative number. However, the slope m=-5 will be more steep, because it’s absolute value is greater. Hope this helps!
• Does infinite slope exist? What happens if the slope of the graph of the line that has a number that does not exist?