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Hi, class.
Today we're going to take a look at a concept called 'slope.'
And when you think of the word 'slope,'
oftentimes people think of a mountain.
And so, let's draw a little mountain here
and pretend that this is a mountain
and you happen to be hiking up it.
Now, if you're hiking up it,
you want to think of what's happening to your position in space.
And you are, actually, as you're hiking along this edge,
moving in towards the center of the mountain
at the same time you're moving up the mountain.
So, as you walk along the slope of the side of the mountain,
you're moving in towards the middle,
we're going to call this your 'run,'
and at the same time you're also moving up the mountain,
which we're going to call your 'rise.'
Slope is actually the ratio
of the 'rise' over the 'run.' [ Slope = rise/run ]
Now, when you're going uphill, like you are here,
and you can tell you're going uphill
by going across the page the direction you read,
and as you go the direction you read,
if it looks like the line's going uphill,
then you're going uphill.
But as you're going uphill,
you'll notice that your run is forward.
When your run is forward, we think of that as a 'positive run.'
And run is in the bottom of the fraction.
Your rise is actually going up, also.
And when you're going up, you think of that as a 'positive rise.'
So we have a positive run and a positive rise.
Positive divided by a positive is positive.
So, when you're going uphill, your slope is positive.
Now, this is going to be helpful when we graph lines,
because if you have a positive slope,
one way to give a little visual check
that you might have done everything okay
is if you're going across the page the direction you read,
if your slope was positive
and it looks like the graph is going uphill,
then you probably did a few things correct.
So, now, let's take a look at what happens
when you get to the other side of the mountain
and you start going downhill.
Now, when you're going downhill, you are dropping,
which you can think of as a negative rise.
But you're going forward still, which in this case is outwards,
away from the center of the mountain,
and forward is still a positive run.
So, downhill, you've got a negative rise over a positive run.
And negative divided by a positive is negative. [- rise/+run = -]
So, we have a slope that is negative
when you're going downhill.
That means when we are graphing lines,
as you go across the paper the direction you read,
if you had a negative slope,
it should look like you're going downhill,
which is a nice little check for you later on.
Now, that gives you kind of an idea of what slope is.
Let's kind of think of what it is in terms of real life.
And we live near a mountain called Mt. Hood.
And as you're driving home on that mountain from Mt. Hood,
it says that you have a 6% downgrade.
That means that for every...
since it's a downgrade, you're dropping,
and for every 6 feet you move forward,
er, every 100 feet you move forward,
you're dropping 6 feet, which is why it's so steep.
Another spot where slope is used is if you look at your house,
your roof in most cases slopes, unless you have a flat roof.
And you can actually find the pitch of your roof
by taking and going inwards from your eaves 12 inches
and then measuring the distance you go up
to hit the edge of your roof.
I actually had to do this in order to figure out
the pitch of my roof to determine how much material I needed
in order to re-do the roof on my house.
And as I went in 12 inches, I ended up going up 5 inches,
so that meant the slope of my roof was five twelfths [ 5/12 ]
which meant that I had a 5/12 pitch on my roof.
And you can go around your entire house and measure each one.
Sometimes your slope changes a little bit.
But on many houses,
it ends up staying the same all the way around.
So, those are some spots where slope is used in real life.
Now, let's take a look at what happens
when you look at slope on a coordinate plane.
So let's make a coordinate plane.
And we're going to go ahead
and plot a line on our coordinate plane.
And that line is going to go through 2 specific points.
It's going to go through the point (2,1)
and it's also going to go through the point (7,4)
which is going to be right here.
Now, we're going to graph that line through these 2 points.
And we're going to try and find the slope of this line.
Now, if you have graph paper and you have them plotted
or you draw a coordinate plane like you see here,
you can actually just count across
for your rise and for your run.
And to travel from this point to this point,
if you follow the grid marks on the graph paper,
you would go from here up to here
and then you would move over to here.
Now, from here up to here,
you went from the level of the 1 up to the level of the 4
which was 1, 2, 3 spaces.
Which meant the rise was 3.
Now, to run over, we moved from here
going from 1, 2, 3, 4, 5.
So, the run was 5,
and that's counting on the little grid marks
if you have graph paper.
Now remember, slope is rise over run,
which in this case ended up
being 3 over 5. [ Slope = rise/run = 3/5 ]
Now, let's say that you don't really like drawing a picture
every single time and counting every single time.
It would be nice to have a formula.
So, let's take a look at how we came up with this 3 and this 5
and if there's a shorter way.
Now, to do that, we're going to label these points.
This first point here, (2,1) has an x and a y.
I'm going to label the x with a little subscript 1
and the y with a little subscript 1.
The little subscripts are just signifying
that's the first point.
So, it's x subscript 1, y subscript 1,
to mean that that's the first point. [ (x1, y1) ]
This other point here is my second point,
so it's going to have, since this is your second x,
it's going to be an x with a subscript 2.
And since this is your second y,
it's going to be a y with a subscript 2,
just signifying that
this is the x and the y from the second point. [ (x2, y2) ]
Now, if we look at this 3, this 3 could also be found
by taking this 4 minus the 1. [ y = 4 - 1 = 3 ]
Now, notice the 4 is the y from the second point [ 4 = y2 ]
And the 1 is the y from the first point. [ 1 = y1 ]
So, our rise could be found by doing 4 minus 1
which is y2 minus y1,
which also gives us our 3. [ 4 - 1 = y2 - y1 = 3 ]
Now, the run, which was on the bottom, was 5.
And 5 can also be found by taking this 7 here
and subtracting from it this 2 here. [ x = 7 - 2 = 5 ]
7 minus 2 gives us this distance here,
and the 7 is the second x [ 7 = x2 ]
and the 2 is the first x. [ 2 = x1 ]
So this would be 7 minus 2 which also gives us the 5
for our run on the denominator.
7 was x2 and 2 was x1. [ 7 - 2 = x2 - x1 = 5 ]
So, x2 minus x1 would also be the run.
So, what we have is that the slope,
which is rise over run, [ Slope = rise/run ]
is also y subscript 2 minus y subscript 1 [ y2 - y1 ]
over x subscript 2 minus x subscript 1. [ /x2 - x1 ]
And the letter that is often given the slope is m.
So, this right here that we just came up with
is called the slope formula. [ m = y2 - y1 / x2 - x1 ]
So if you were asked to find the slope of a line,
you could either graph it and count off,
or you could just plug in to this formula.
Let's do a problem where we use this formula to find the slope,
and then we draw a picture to double check it.
So let's say you're told to find the slope of the line
through the point (-1,3) and (2,-4).
Well, our formula we just came up with
was that slope was the second y minus the first y
over the second x minus the first x. [ m = y2 - y1 / x2 - x1 ]
Let's just label
that this one's (-1,3) going to be the first x and the first y.
And this one (2,-4)
will be the second x and this will be the second y.
Labeling them like that and just determining
I'm going to make this one here my first one
and that my second one
helps people out quite a bit. [ (-1,3) (2-4) ]
Now, once we've done that, we go through our formula
and plug in the appropriate values.
y2 we labeled as negative 4,
so I'm going to put a negative 4 here in that spot.
Minus the y1 I labeled as 3.
So I'm going to put a 3 there. [ m = (-4) - (3)/ x2 - x1 ]
x2 is this here, because I labeled that x2
so I'm going to put a 2 there. [ m = -4 - 3/ 2 - x1 ]
Minus x1 I labeled here,
so that's going to be negative 1. [ m = -4 - 3/ 2 - (-1) ]
[ m = y2 - y1/ x2 - x1 = -4 - 3/ 2 - (-1) ] Now I just calculate.
On the bottom here, [ / 2 - (-1) ] I have a double negative,
so that's going to be a positive. [ / 2 + 1 ]
So on the top I get negative 7, [ -4 - 3 = -7 ]
and on the bottom, 2 plus 1 is 3. [ 2 + 1 = 3 ]
So that should be my slope. [ m = -7/3 ]
Let's graph it and count and double check.
So I'm going to make a coordinate plane
and label my x and my y.
And I'm going to go ahead and plot my points.
I have (-1,3) right here,
and I have (2,-4) right here.
If I count,
(oops),
from one point to the next,
following the grid marks, here I'm going to go
down to there
and then I'm going to go over to here.
Going down, I had to go down 1, 2, 3, 4, 5, 6, 7 spaces,
so that's a negative 7 for my rise.
And then going over, I had to go over 1, 2, 3 spaces,
so that's a 3 for my run,
which gives me a slope of negative 7 over 3. [ -7/3 ]
So you can see that we got the same answer
going at the problem both directions, using the formula one way
and drawing the graph and counting the other way.