Main content

## 8th grade

### Course: 8th grade > Unit 3

Lesson 4: Slope- Intro to slope
- Intro to slope
- Slope formula
- Slope & direction of a line
- Positive & negative slope
- Worked example: slope from graph
- Slope from graph
- Slope of a line: negative slope
- Worked example: slope from two points
- Slope from two points
- Slope from equation
- Converting to slope-intercept form
- Slope from equation
- Slope of a horizontal line
- Slope review

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Positive & negative slope

CCSS.Math: , , ,

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?(24 votes)
**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!(69 votes)

- 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(31 votes)

- Does the slope line only have to be in the NE direction? Or can it be in the opposite direction, like NW?(2 votes)
- 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 anthat**increasing slope****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**too.*larger*

★**Negative slopes**have a, so they**decreasing slope****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!(16 votes)

- Did you purposefully make lines 1, 2, and 4 (pink, blue, and orange) converge on the same point?(6 votes)
- 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?(5 votes)
- 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(2 votes)

- can you get a slope with a decimal point in it(3 votes)
- 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.(3 votes)

- 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)
- 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)

- can there be a slope value = e(2 votes)
- There could be, but it is hard to graph because it is irrational (will never have a point (x,y) where x and y are both integers).(4 votes)

- 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?(1 vote)
- 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.(6 votes)

- which slope is greater, m=4 or m=-5(2 votes)
- 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!(4 votes)

## 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.