Video Transcript
In this video we’re gonna explore
the general form of the equation of a straight line function: 𝑦 equals 𝑚𝑥 plus
𝑏. Now depending on where you live,
you may have seen this written as 𝑦 equals 𝑚𝑥 plus 𝑐. Well, it means just the same
thing. But some bright spark thought it’d
be a good idea to use the letter “𝑐” instead of the letter “𝑏” to represent the
value of the coordinate of the point that cuts the 𝑦-axis. Now you think that’ll it make it
easier to remember which value tells you where the line cuts the 𝑦-axis — 𝑐 stands
for cut. But after many years of teaching in
places where they use 𝑦 equals 𝑚𝑥 plus 𝑐 rather than 𝑦 equals 𝑚𝑥 plus 𝑏,
believe me it doesn’t work. You might as well just use 𝑦
equals 𝑚𝑥 plus 𝑏. Anyway whichever version of the
formula you use, this is the big idea we’re gonna be looking at in this video.
So let’s look at some graphs
first. Let’s plot 𝑦 equals 𝑥. Well 𝑦 equals 𝑥 could also be
written as 𝑦 equals one 𝑥 or 𝑦 equals one times 𝑥. So whatever 𝑥-coordinate we give,
we just multiply it by one and that will also be the 𝑦-coordinate. So we’ve got a series of points
that have exactly the same 𝑥- and 𝑦-coordinates. So for example, if my 𝑥-coordinate
is zero, my 𝑦-coordinate would also be zero; if my 𝑥-coordinate is five, my
𝑦-coordinate is also five; or if my 𝑥-coordinate is negative seven, the
𝑦-coordinate would also be negative seven. And that leads us to a whole series
of points and loads of points in between them that look like this on the graph. And when I join them up, they look
like this, giving us the graph of the line 𝑦 equals 𝑥.
Now let’s plot the line 𝑦 equals
two 𝑥. And for this one, we had to take
the 𝑥-coordinate and times it by two to get the 𝑦-coordinate. So if our 𝑥-coordinate was zero,
we’d multiply that by two and we’d also get zero. And if the 𝑥-coordinate was three,
we’d multiply that by two and get six and so on. So our graph would look like
this.
And in the same way, we could plot
𝑦 equals three 𝑥. Now already I think we’ll plot two-
we’ll plot 𝑦 equals a half 𝑥 and 𝑦 equals zero 𝑥. Well clearly 𝑦 equals zero 𝑥 is
just 𝑦 equals zero. So the 𝑦-coordinate is always
gonna be zero in that case. So there’s 𝑦 equals a half 𝑥 and
we can see that the 𝑥-coordinate is ten at this point here and our 𝑦-coordinate is
five. So the 𝑦-coordinate is half of the
𝑥-coordinate and that’s true for all of the coordinates on that line. And there’s 𝑦 equals zero 𝑥,
where the 𝑦-coordinate is always zero.
So now we’ve plotted all those
lines together. What you should notice is that the
multiplier of the 𝑥 tells us how steep that line is gonna be. We got a series of lines which
start off horizontal and get steeper and steeper and steeper as the multiplier of 𝑥
gets bigger. Let’s pick a point in one of those
lines and increase our 𝑥-coordinate by one. Now when we do that, the
corresponding 𝑦-coordinate back on the line has gone up by a half and look that’s
the multiplier of 𝑥. When I look at the line 𝑦 equals
one 𝑥 if I take a point on the line, increase the 𝑥-coordinate by one, the
corresponding 𝑦-coordinate if I move back to the line has also gone up by one. If I do the same on the line 𝑦
equals two 𝑥, increase the 𝑥-coordinate by one, the corresponding 𝑦-coordinate to
get back to the line goes up by two and that’s the multiplier of 𝑥.
So for any straight line in this 𝑦
equals 𝑚𝑥 plus 𝑏 format, so 𝑦 equals a number times 𝑥 plus or minus another
number; I mean in our case the numbers are plus zero, so that 𝑏 or 𝑐 number on the
end is zero, any line in that format the multiplier of 𝑥 tells you about the slope
of the line — how steep it is or how shallow it is: is it horizontal? or is it
getting more and more vertical? But the number itself specifically
tells you how much will the 𝑦-coordinate change if I increase my 𝑥-coordinate by
one at any point on that line.
Right, let’s go through that
exercise again. But this time our multipliers of 𝑥
are going to be negative numbers. So we’re gonna do zero 𝑥, negative
a half 𝑥, negative one 𝑥, negative two 𝑥, and negative three 𝑥. So 𝑦 equals zero 𝑥 is still this
horizontal line. And for 𝑦 equals negative a half
𝑥, the 𝑦-coordinate is negative a half times the 𝑥-coordinate. So for example, when the
𝑥-coordinate is five, the 𝑦-coordinate is negative a half times that. So it’s negative two point
five. So you can see this point on the
line here.
Now we can also plot 𝑦 equals
negative one 𝑥 or just negative 𝑥, and 𝑦 equals negative two 𝑥, and 𝑦 equals
negative three 𝑥. Now again 𝑦 equals zero 𝑥. We said it was a horizontal line
and you’ll see that as this number gets bigger and more negative the line gets
steeper. But instead of going from the
bottom left to the top right, going uphill, it’s going in the other direction. It’s going from the top left to the
bottom right; it’s going where we would call downhill. So the number on the multiplier of
𝑥 tells us about the slope of that line, but the sign of that number also gives us
some information about the slope. If it’s negative, it’s going in
this downhill direction top left to bottom right. If it was positive, it’s going in
this uphill direction as we increase 𝑥 from the bottom left up to the top
right.
Now because every time I increase
my 𝑥-coordinate by one on a given line, the 𝑦-coordinate always changes by the
same amount; that’s what makes it a straight line. So on the line 𝑦 equals negative
three 𝑥 if I start up here and I increase the 𝑥-coordinate by one, to get back
into that line I’ve got to decrease the 𝑦-coordinate by three. And that same thing is true no
matter whereabouts on the straight line I start. And looking at 𝑦 equals negative a
half 𝑥, the same thing is true. Every time no matter wherever I
start on that line if I increase my 𝑥-coordinate by one, the 𝑦-coordinate
decreases by the same amount — negative a half in this case. So whether it’s from here or it’s
here or here or even over here, the slope of that line is always the same,
everywhere on the line.
Well now we’re gonna plot some
different lines: 𝑦 equals 𝑥 plus zero, 𝑦 equals 𝑥 plus one, 𝑥 plus two, 𝑥 plus
three, 𝑥 plus four, and 𝑥 plus five. And you can do this using some
software or you could do it using tables of values. But I’m going to do it for you
here; what we do is we take our 𝑥-coordinate and we add something to it. Now depending on which equation
we’re using, we’ll either add one or nothing and two or three or four or five. So if I take my 𝑥-coordinate and
add nothing to it to get my 𝑦-coordinate, it means all my 𝑥- and 𝑦-coordinates
are the same. So this is the line that we
get. If I take my 𝑥-coordinate and add
one to it all the time, then this is the line I get. Look when my 𝑥-coordinate is zero,
zero plus one is one, so my 𝑦-coordinate will be one; so this is the point I’m
gonna get. If my 𝑥-coordinate is four, four
plus one is five, so five will be the 𝑦-coordinate I get. So that’s the line 𝑦 equals 𝑥
plus one. Now I’m gonna plot 𝑥 plus two, 𝑥
plus three, 𝑥 plus four, and 𝑥 plus five. And here are the lines that I
get. Now this isn’t massively surprising
because look they’ve all got the same multiplier of 𝑥. We didn’t have anything in front of
it; so it’s all one times 𝑥. So the slope of those lines should
all be the same. Every time I increase my
𝑥-coordinate by one, the 𝑦-coordinate goes up by one. They’re parallel lines because
they’ve all got the same slope of one.
But now let’s look at this other
thing: the plus 𝑏. We’re adding zero, one, two, three,
four, and five. 𝑦 equals 𝑥 plus zero cuts the
𝑦-axis here at zero, 𝑦 equals 𝑥 plus one cuts the 𝑦-axis here at one, 𝑦 equals
𝑥 plus two cuts the 𝑦-axis here at two, and no surprises 𝑦 equals 𝑥 plus three,
four, and five cuts the 𝑦-axis here at three, four, and five. So this second term here, the plus
number on its own or the minus number on its own, will tell us where that line is
gonna cut the 𝑦-axis.
Okay let’s take a look at some
lines then when we are subtracting a number rather than adding a number on the
end. So 𝑦 equals 𝑥 plus zero, 𝑥 minus
one, 𝑥 minus two, and so on down to 𝑥 minus five. So this is what those lines look
like. Again remember that they all mean
one times 𝑥. So the slope of the line is
one. They’ve all got the same slope;
therefore, they’re parallel. And the thing that’s different
about all of those lines is the number we’re adding on the end and that tells us
whereabouts it cuts the 𝑦-axis.
So now we know how equations of
straight lines work; we know the rules. We can easily plot linear graphs
from the equations without having to make a table of values first. So we just need first of all to
plot a 𝑦-intercept; it’s minus three, so that’s here. And then as I increase my
𝑥-coordinate, my 𝑦-coordinate is increasing by two because I’ve got a positive two
in front of the 𝑥. So as I go increase my
𝑥-coordinate by one, my 𝑦-coordinate is gonna go up by two, increase the
𝑥-coordinate by one, the 𝑦-coordinate goes up by two. And if I start decreasing my
𝑥-coordinate by one, the 𝑦-coordinate is gonna do the opposite thing; it’s gonna
come down by two. So as I decrease my 𝑥-coordinate
by one, the 𝑦-coordinate comes down by two, decrease by one, comes down by two, and
so on. Then I just have to join all these
up. And here’s my line. So remember the number in front of
the 𝑥 tells us the slope of the line. If it’s positive, it’s an uphill
line. If it was negative, it will be a
downhill line. So that’s a little check that you
need to sort of learn and remember and the number on the end tells you where it cuts
the 𝑦-axis. So negative three; it’s cutting the
𝑦-axis at negative three.
Now let’s plot 𝑦 equals one and a
half 𝑥 plus two. So here is where it cuts the
𝑦-axis. And this multiplier of 𝑥 tells us
the slope of the graph. Every time I increase the
𝑥-coordinate by one, the 𝑦-coordinate goes up by one and a half. Or to make it easier with whole
numbers doubling those, if I increase the 𝑥-coordinate by twos — that’s twice as
many, then the 𝑦-coordinate goes up by three — that’s twice as many. So increasing the 𝑥-coordinate by
two, the 𝑦-coordinate goes up by three. Increasing the 𝑥-coordinate by
two, the 𝑦-coordinate goes up by three, and so on. And likewise coming back, if I
decrease the 𝑥-coordinate by two, the 𝑦-coordinate is gonna go down by three,
decrease the 𝑥-coordinate by two, the 𝑦-coordinate goes in the opposite direction
down by three, and so on. And joining up the points, there’s
our line. The slope was a positive number;
the multiplier of 𝑥 was a positive number. So we know it should be an uphill
line, which it is; so that’s good. And it cuts the 𝑦-axis at positive
two, which it does; so that’s good.
Now let’s plot 𝑦 equals negative
𝑥 plus five. So looking at negative 𝑥, remember
that means a negative one 𝑥. So our slope is negative one. Every time I increase my
𝑥-coordinate by one, my 𝑦-coordinate goes down by one. Now the number on its own is
positive five. So this cuts the 𝑦-axis at
positive five; the intercept is five. So let’s plot that intercept
there. And then every time I increase the
𝑥-coordinate by one, the 𝑦-coordinate goes down by one, increase the 𝑥 by one,
the 𝑦-coordinate goes down by one. And let’s carry on with that
pattern to get these points and then go backwards. Every time I decrease my
𝑥-coordinate by one, the 𝑦-coordinate is gonna go up by one; the opposite of the
negative side, it’s gonna go up by one. Decrease my 𝑥-coordinate by one,
𝑦-coordinate goes up by one, and so on. And joining up those points,
there’s my line. Remember cuts the 𝑦-axis at five
yup. And it’s a negative slope, so
that’s a downhill slope. Every time I increase my
𝑥-coordinate by one, the 𝑦-coordinate decreases by one; that’s negative one 𝑥,
which matches my graph.
Now let’s plot 𝑦 plus 𝑥 equals
two. Well we’ve a slight problem here
because this isn’t quite in this right format. It’s not in our 𝑦 equals something
times 𝑥 plus another number format. So what I’m gonna do is I’m going
to subtract 𝑥 from both sides of my equation, which gives me 𝑦 plus 𝑥 take away
𝑥 on the left-hand side and two take away 𝑥 on the right-hand side. So if I start off with positive 𝑥
and then I take away 𝑥, those two things are gonna cancel each other out. So I’ve just got 𝑦 on the
left-hand side of my equation. Now on the right-hand side, I’ve
got two take away 𝑥; now it doesn’t matter whether I say two take away 𝑥 or if I
start off with negative 𝑥 and then I add two onto it. Remember this two on its own here
is really a positive two. And in this format, it’s really
easy to recognize that the negative 𝑥 means the negative one 𝑥. So the slope is negative one. And that plus two on its own here
tells us that we’re cutting the 𝑦-axis at positive two. So I can put that on the graph. Here we go, cut the 𝑦-axis at
positive two and then every time I increase my 𝑥-coordinate by one, the 𝑦
coordinate decreases by one. So these are the points you’re
gonna generate. And when you join them up that’s
what they’re gonna look like.
Now it’s also worth noting at this
point I’ve been drawing all these little orange lines and showing you know increase
𝑥 by one the 𝑦-coordinate goes down by one, you don’t actually need to be doing
those. So you only need to be plotting the
points. So I’m just putting them on the
page so you can see them. It makes it nice and clear what
we’re doing — how we’re counting this up, but you wouldn’t normally draw those
orange lines in when you’re plotting these graphs. Now the key learning point for this
example was the fact that sometimes we have to rearrange our equation in order to
get that 𝑦 equals 𝑚𝑥 plus 𝑏 format, which makes it much easier to plot.
Let’s do another couple of examples
then.
So for this one, I’ve got 𝑥 plus
𝑦 plus three equals zero. I’m gonna have to do a bit more
rearranging. So first of all, I’m gonna subtract
𝑥 from both sides. And when I do that, I’ve got 𝑥 and
I’m taking away 𝑥 on the left-hand side, so I can cancel those two out. And on the right-hand side, I got
zero take away 𝑥. Well that is just negative 𝑥. So I’ve got 𝑦 plus three is equal
to negative 𝑥, still not quite the right format. So I need to take away three from
both sides. And when I do that over on the
left-hand side, I’ve got 𝑦 add three take away three. So if I have to do three take away
three, that’s nothing; so these two terms over here cancel out. So I’ve got 𝑦 is equal to negative
𝑥 take away three. Great, that’s now in my 𝑦 equals
𝑚𝑥 plus 𝑏 format. And negative 𝑥 is the same as
negative one 𝑥, so our slope is negative one. So cutting the 𝑦-axis at negative
three looks like that. And without doing all the ziggity
zaggity lines all over the place, we can see that increasing our 𝑥-coordinate
decreases our 𝑦-coordinate by one, leaves with these points which look like this
when I join them up. So that’s the equation 𝑥 plus 𝑦
plus three equals zero, which I rearranged to 𝑦 equals negative 𝑥 take away three
in order to quickly be able to see what the slope was, negative one, and where the
𝑦-intercept was, negative three.
So the big lesson of this example
is that sometimes we need to rearrange our equation in order to get it into that 𝑦
equals 𝑚𝑥 plus 𝑏 format so that we can easily work out the slope and the
intercept.
So to summarize then, 𝑦 equals
𝑚𝑥 plus 𝑏 or 𝑚𝑥 plus 𝑐 is the general form of the straight line equation. The multiplier 𝑥 is the slope,
which means that when I increase my 𝑥-coordinate by one, the 𝑦-coordinate will
increase by whatever that 𝑚 value is — whether it’s positive or whether it’s
negative. And the 𝑏 value tells us the
𝑦-coordinate of the point on the line that cuts the 𝑦-axis. If the 𝑚 value was positive, we’ll
have an uphill line that looks like this. And if it was negative, we’ll have
a downhill line like that. Sometimes we need to rearrange the
equation to get the 𝑦 equals 𝑚𝑥 plus 𝑏 format and hence the slope and where it
cuts the 𝑦-axis. And lastly remember not all
equations rearrange into the 𝑦 equals 𝑚𝑥 plus 𝑏 format. Not all functions represent
straight line graphs.
