Hi everyone it's Jenni, with the How and Why of Mathematics, and today I'm going to
talk about how to find the y-intercept of a straight line if you just know the
coordinates of some points on the line. And I don't want you to see this just as a
procedure to follow but as an opportunity to improve your problem-solving
skills, because I want you to be able to figure this out from what you already
know. Ok, so let's say you're trying to work out the y-intercept of the straight
line through the points (-4,1) and (5,4) and I'm assuming that you already know
how to work out slope. If you don't then go back to my previous video that I did
on that. Let's say we've already worked out that it's 1 over 3 and we just
need the y-intercept which is the c in y = mx + c, so you can work out
the equation of the line from the y-intercept or vice versa; you can work
out the y-intercept if you already know the equation. And graphically speaking,
the y-intercept is just the point where the line crosses the y axis. So that's
the y-axis there; that's the x-axis. Y-intercept is just the coordinate there.
Now you might like to pause this video and see if you can actually figure out
how to work out the y-intercept yourself and then come back. Alright now there's a
lot of ways that you could figure this out. I'm gonna start with my favorite
method which is just to say: okay so far we know that y is 1/3 x plus
something, we don't know what that is yet but so far we have that. And then we can
just say: well, these points are on that line, right? So they have to satisfy this
equation, so we can just pick one of them, we can go with the (-4,1) and
sub that in so y is 1 and x is -4. So we know that this equation has to
be true. And then we've just got an
equation that only involves c and then we can solve that for c. So that's
-4/3 + c. We wanna move that over the other side so add 4/3
to both sides and then you've got 1 + 4/3 so c is 1 + 4/3, and 1 is 3/3
so if you add those up you'll get 7/3 and there it is; that's the
y-intercept. So if you were asked to find the equation of the line, you would then
write y = 1/3 x + 7/3. Another way that you could do it is you
could think about it graphically, so on our graph, we can put the (-4,1)
there and (5,4) is up here. This isn't the best drawn graph but that's ok, doesn't
really matter. Now let's just focus on the (-4,1) and we'll try to work
out the coordinates of this point. And we already know that the slope is 1 on 3 so
we can use that because we can draw in a little triangle here where we know that
that divided by that is 1 over 3 and we can work that out because we know that
the x-coordinate here is negative 4 so the distance from there to there is 4
and if that divided by that is 1/3 then this has to be 4 on 3.
Well that's really good now because we know where this is: that's at 1 and then
this is just 4/3 up from 1, so therefore the y intercept has to be 1
+ 4/3 which will give us 7/3 like before. Now let's just work out
what we did to work this out so that we can apply it to other coordinates and
maybe even without having to draw the graph. So we had the the y coordinate, so
let's call that y1 and then we had the slope times the x coordinate, except that
we made it the opposite sign because the slope times the x coordinate in this case would
have been -4/3 and we actually added 4/3
so it's going to be y1 minus m x1. That's how we worked that out. So that's a
formula you can memorize and then you can apply it to other coordinates. Yet
another way that you could figure it out is you could use the formula for slope.
So we know slope is y2 minus y1 over x2 minus x1 but instead of using two known
points on the line, we could use one of the points and one arbitrary point (x,y)
because no matter what point you pick on the line it has to have a slope of 1/3
when you calculate it, you know, between these two points, so we can use this as
our x2 and y2 and this is our x1 and y1 and plug it into this equation. So you
would have... we know the slope's 1/3 and then I'm going to use our arbitrary
point for the y2 and x2 and then for these ones you can pick either point;
I'll use the (-4,1) so the y1 is 1 and the x1 is -4 and if we're
subtracting -4 then we're adding 4. Now you've got this equation that you
can solve for y, so let's multiply both sides by the x + 4, so we get a
third x plus 4 equals y minus 1 and then we can expand that and move the 1 over
and again we're gonna get Y equals a third x plus 7/3 like we did before.
And then if you were just asked to find the y intercept you would say, okay it's
that number because it's always the one that's the constant term, the one that's
not next to the x. And... all right let's turn this into like a
formula that we can use for other points. So what we did was we had this with just
y and x there, so not a particular point on the line but just
standing for every point on the line basically, and then we multiplied both
sides by the x - x1, so I'm going to flip this around as well, so we got y
minus y1 equals m x minus x1 and then solve that for y. So this is a formula
that's often used for finding y-intercept as well, so you can memorize
that if you like or you can just, you know, figure it out from first principles
again each time. Now another way that you can work out that same equation is
something which I think you probably haven't covered yet but I thought I'd
mention it anyway. What we could do is imagine a line that passes through
the origin and has that slope, one on three, so that would look something like
that and that would have the equation y equals one on three x. You can tell
that that passes through the origin because if you plug in x equals zero you
get y equals zero, so it has to pass through the point (0,0). Now we don't want
this though what we want is for it to pass through (-4,1) so we
want it to... we wanna have that line which is the same as this line but
shifted across. Now there's a way to shift graphs which works for any kind
of graph not just straight lines which is to replace x and y with x minus the x
coordinate here and y minus the y coordinate here so you would take this
original equation which is for that line, replace the y with y minus the y
coordinate which was one, and replace the x with x minus the x coordinate here, which
was -4 so x minus negative 4 is x plus 4, so that works for shifting any
graph like circles, parabolas, hyperbolas, whatever. If you replace x and
y with x-x1 and y-y1 that'll shift it to those coordinates.
Alright I'll go through an example so you can see how to apply this. So let's
say we're trying to work out the equation of the straight line which
passes through the points (5,0) and (3,-4) and let's say we've already worked
out that the slope is 2, again go to my previous video if you've forgotten how
to do that. So I'll start with my favorite method, which
is to say: so far we know that y equals 2x plus c because we know that m is 2 and
then we can sub one of these into here, so I might pick the (5,0) because that's
gonna be easiest. That'll give us 0 equals two times five plus c and then solve
that for c; that'll give you c equals negative 10, so that's your y-intercept
and then if you were asked to find the equation you would say y equals 2x minus
10. So that's my favorite method. You could also use your y - y1 equals
m (x - x1), so we go y minus one of the y coordinates, we could use the 0 for
instance, equals... the m is 2, x minus 5. It doesn't matter which point you pick;
you can pick either one, doesn't matter. And then that simplifies to y equals 2x
minus 10 so you can do it that way or you can also just memorize that c is y1
minus m x1 so in that case that would give you... we
can use that as our y1 and then minus 2 times 5 which is negative 10 and then if
you were asked to find the equation, of course you can't just leave it there; you have to say
y = 2x minus 10, and then what I would do after that, just to check that
it's correct, is check that both of these points satisfy the equation, so you can
do that in your head, you don't have to do it in your working, but just check ok y is
negative 4 and then on the right you've got 2 times the 3 which is 6 minus 10
which makes negative 4. That's good! So the points have to satisfy that equation
and check it with the other one too, although that one's probably right
because that's the one that we used to work it out.
All right that's it for y intercept, so there's a link in the
description below for where you can get practice questions on y-intercept so
make sure you check that out if you don't want to forget this and be sure to
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