### Video Transcript

In this video weโre gonna explore
the general form of the equation of a straight line function: ๐ฆ equals ๐๐ฅ plus
๐. Now depending on where you live,
you may have seen this written as ๐ฆ equals ๐๐ฅ plus ๐. Well, it means just the same
thing. But some bright spark thought itโd
be a good idea to use the letter โ๐โ instead of the letter โ๐โ to represent the
value of the coordinate of the point that cuts the ๐ฆ-axis. Now you think thatโll it make it
easier to remember which value tells you where the line cuts the ๐ฆ-axis โ ๐ stands
for cut. But after many years of teaching in
places where they use ๐ฆ equals ๐๐ฅ plus ๐ rather than ๐ฆ equals ๐๐ฅ plus ๐,
believe me it doesnโt work. You might as well just use ๐ฆ
equals ๐๐ฅ plus ๐. Anyway whichever version of the
formula you use, this is the big idea weโre gonna be looking at in this video.

So letโs look at some graphs
first. Letโs plot ๐ฆ equals ๐ฅ. Well ๐ฆ equals ๐ฅ could also be
written as ๐ฆ equals one ๐ฅ or ๐ฆ equals one times ๐ฅ. So whatever ๐ฅ-coordinate we give,
we just multiply it by one and that will also be the ๐ฆ-coordinate. So weโve got a series of points
that have exactly the same ๐ฅ- and ๐ฆ-coordinates. So for example, if my ๐ฅ-coordinate
is zero, my ๐ฆ-coordinate would also be zero; if my ๐ฅ-coordinate is five, my
๐ฆ-coordinate is also five; or if my ๐ฅ-coordinate is negative seven, the
๐ฆ-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 ๐ฆ equals ๐ฅ.

Now letโs plot the line ๐ฆ equals
two ๐ฅ. And for this one, we had to take
the ๐ฅ-coordinate and times it by two to get the ๐ฆ-coordinate. So if our ๐ฅ-coordinate was zero,
weโd multiply that by two and weโd also get zero. And if the ๐ฅ-coordinate was three,
weโd 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
๐ฆ equals three ๐ฅ. Now already I think weโll plot two-
weโll plot ๐ฆ equals a half ๐ฅ and ๐ฆ equals zero ๐ฅ. Well clearly ๐ฆ equals zero ๐ฅ is
just ๐ฆ equals zero. So the ๐ฆ-coordinate is always
gonna be zero in that case. So thereโs ๐ฆ equals a half ๐ฅ and
we can see that the ๐ฅ-coordinate is ten at this point here and our ๐ฆ-coordinate is
five. So the ๐ฆ-coordinate is half of the
๐ฅ-coordinate and thatโs true for all of the coordinates on that line. And thereโs ๐ฆ equals zero ๐ฅ,
where the ๐ฆ-coordinate is always zero.

So now weโve plotted all those
lines together. What you should notice is that the
multiplier of the ๐ฅ 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 ๐ฅ
gets bigger. Letโs pick a point in one of those
lines and increase our ๐ฅ-coordinate by one. Now when we do that, the
corresponding ๐ฆ-coordinate back on the line has gone up by a half and look thatโs
the multiplier of ๐ฅ. When I look at the line ๐ฆ equals
one ๐ฅ if I take a point on the line, increase the ๐ฅ-coordinate by one, the
corresponding ๐ฆ-coordinate if I move back to the line has also gone up by one. If I do the same on the line ๐ฆ
equals two ๐ฅ, increase the ๐ฅ-coordinate by one, the corresponding ๐ฆ-coordinate to
get back to the line goes up by two and thatโs the multiplier of ๐ฅ.

So for any straight line in this ๐ฆ
equals ๐๐ฅ plus ๐ format, so ๐ฆ equals a number times ๐ฅ plus or minus another
number; I mean in our case the numbers are plus zero, so that ๐ or ๐ number on the
end is zero, any line in that format the multiplier of ๐ฅ tells you about the slope
of the line โ 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 ๐ฆ-coordinate change if I increase my ๐ฅ-coordinate by
one at any point on that line.

Right, letโs go through that
exercise again. But this time our multipliers of ๐ฅ
are going to be negative numbers. So weโre gonna do zero ๐ฅ, negative
a half ๐ฅ, negative one ๐ฅ, negative two ๐ฅ, and negative three ๐ฅ. So ๐ฆ equals zero ๐ฅ is still this
horizontal line. And for ๐ฆ equals negative a half
๐ฅ, the ๐ฆ-coordinate is negative a half times the ๐ฅ-coordinate. So for example, when the
๐ฅ-coordinate is five, the ๐ฆ-coordinate is negative a half times that. So itโs negative two point
five. So you can see this point on the
line here.

Now we can also plot ๐ฆ equals
negative one ๐ฅ or just negative ๐ฅ, and ๐ฆ equals negative two ๐ฅ, and ๐ฆ equals
negative three ๐ฅ. Now again ๐ฆ equals zero ๐ฅ. We said it was a horizontal line
and youโll 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, itโs going in the other direction. Itโs going from the top left to the
bottom right; itโs going where we would call downhill. So the number on the multiplier of
๐ฅ tells us about the slope of that line, but the sign of that number also gives us
some information about the slope. If itโs negative, itโs going in
this downhill direction top left to bottom right. If it was positive, itโs going in
this uphill direction as we increase ๐ฅ from the bottom left up to the top
right.

Now because every time I increase
my ๐ฅ-coordinate by one on a given line, the ๐ฆ-coordinate always changes by the
same amount; thatโs what makes it a straight line. So on the line ๐ฆ equals negative
three ๐ฅ if I start up here and I increase the ๐ฅ-coordinate by one, to get back
into that line Iโve got to decrease the ๐ฆ-coordinate by three. And that same thing is true no
matter whereabouts on the straight line I start. And looking at ๐ฆ equals negative a
half ๐ฅ, the same thing is true. Every time no matter wherever I
start on that line if I increase my ๐ฅ-coordinate by one, the ๐ฆ-coordinate
decreases by the same amount โ negative a half in this case. So whether itโs from here or itโs
here or here or even over here, the slope of that line is always the same,
everywhere on the line.

Well now weโre gonna plot some
different lines: ๐ฆ equals ๐ฅ plus zero, ๐ฆ equals ๐ฅ plus one, ๐ฅ plus two, ๐ฅ plus
three, ๐ฅ plus four, and ๐ฅ plus five. And you can do this using some
software or you could do it using tables of values. But Iโm going to do it for you
here; what we do is we take our ๐ฅ-coordinate and we add something to it. Now depending on which equation
weโre using, weโll either add one or nothing and two or three or four or five. So if I take my ๐ฅ-coordinate and
add nothing to it to get my ๐ฆ-coordinate, it means all my ๐ฅ- and ๐ฆ-coordinates
are the same. So this is the line that we
get. If I take my ๐ฅ-coordinate and add
one to it all the time, then this is the line I get. Look when my ๐ฅ-coordinate is zero,
zero plus one is one, so my ๐ฆ-coordinate will be one; so this is the point Iโm
gonna get. If my ๐ฅ-coordinate is four, four
plus one is five, so five will be the ๐ฆ-coordinate I get. So thatโs the line ๐ฆ equals ๐ฅ
plus one. Now Iโm gonna plot ๐ฅ plus two, ๐ฅ
plus three, ๐ฅ plus four, and ๐ฅ plus five. And here are the lines that I
get. Now this isnโt massively surprising
because look theyโve all got the same multiplier of ๐ฅ. We didnโt have anything in front of
it; so itโs all one times ๐ฅ. So the slope of those lines should
all be the same. Every time I increase my
๐ฅ-coordinate by one, the ๐ฆ-coordinate goes up by one. Theyโre parallel lines because
theyโve all got the same slope of one.

But now letโs look at this other
thing: the plus ๐. Weโre adding zero, one, two, three,
four, and five. ๐ฆ equals ๐ฅ plus zero cuts the
๐ฆ-axis here at zero, ๐ฆ equals ๐ฅ plus one cuts the ๐ฆ-axis here at one, ๐ฆ equals
๐ฅ plus two cuts the ๐ฆ-axis here at two, and no surprises ๐ฆ equals ๐ฅ plus three,
four, and five cuts the ๐ฆ-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 ๐ฆ-axis.

Okay letโs take a look at some
lines then when we are subtracting a number rather than adding a number on the
end. So ๐ฆ equals ๐ฅ plus zero, ๐ฅ minus
one, ๐ฅ minus two, and so on down to ๐ฅ minus five. So this is what those lines look
like. Again remember that they all mean
one times ๐ฅ. So the slope of the line is
one. Theyโve all got the same slope;
therefore, theyโre parallel. And the thing thatโs different
about all of those lines is the number weโre adding on the end and that tells us
whereabouts it cuts the ๐ฆ-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 ๐ฆ-intercept; itโs minus three, so thatโs here. And then as I increase my
๐ฅ-coordinate, my ๐ฆ-coordinate is increasing by two because Iโve got a positive two
in front of the ๐ฅ. So as I go increase my
๐ฅ-coordinate by one, my ๐ฆ-coordinate is gonna go up by two, increase the
๐ฅ-coordinate by one, the ๐ฆ-coordinate goes up by two. And if I start decreasing my
๐ฅ-coordinate by one, the ๐ฆ-coordinate is gonna do the opposite thing; itโs gonna
come down by two. So as I decrease my ๐ฅ-coordinate
by one, the ๐ฆ-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 hereโs my line. So remember the number in front of
the ๐ฅ tells us the slope of the line. If itโs positive, itโs an uphill
line. If it was negative, it will be a
downhill line. So thatโs a little check that you
need to sort of learn and remember and the number on the end tells you where it cuts
the ๐ฆ-axis. So negative three; itโs cutting the
๐ฆ-axis at negative three.

Now letโs plot ๐ฆ equals one and a
half ๐ฅ plus two. So here is where it cuts the
๐ฆ-axis. And this multiplier of ๐ฅ tells us
the slope of the graph. Every time I increase the
๐ฅ-coordinate by one, the ๐ฆ-coordinate goes up by one and a half. Or to make it easier with whole
numbers doubling those, if I increase the ๐ฅ-coordinate by twos โ thatโs twice as
many, then the ๐ฆ-coordinate goes up by three โ thatโs twice as many. So increasing the ๐ฅ-coordinate by
two, the ๐ฆ-coordinate goes up by three. Increasing the ๐ฅ-coordinate by
two, the ๐ฆ-coordinate goes up by three, and so on. And likewise coming back, if I
decrease the ๐ฅ-coordinate by two, the ๐ฆ-coordinate is gonna go down by three,
decrease the ๐ฅ-coordinate by two, the ๐ฆ-coordinate goes in the opposite direction
down by three, and so on. And joining up the points, thereโs
our line. The slope was a positive number;
the multiplier of ๐ฅ was a positive number. So we know it should be an uphill
line, which it is; so thatโs good. And it cuts the ๐ฆ-axis at positive
two, which it does; so thatโs good.

Now letโs plot ๐ฆ equals negative
๐ฅ plus five. So looking at negative ๐ฅ, remember
that means a negative one ๐ฅ. So our slope is negative one. Every time I increase my
๐ฅ-coordinate by one, my ๐ฆ-coordinate goes down by one. Now the number on its own is
positive five. So this cuts the ๐ฆ-axis at
positive five; the intercept is five. So letโs plot that intercept
there. And then every time I increase the
๐ฅ-coordinate by one, the ๐ฆ-coordinate goes down by one, increase the ๐ฅ by one,
the ๐ฆ-coordinate goes down by one. And letโs carry on with that
pattern to get these points and then go backwards. Every time I decrease my
๐ฅ-coordinate by one, the ๐ฆ-coordinate is gonna go up by one; the opposite of the
negative side, itโs gonna go up by one. Decrease my ๐ฅ-coordinate by one,
๐ฆ-coordinate goes up by one, and so on. And joining up those points,
thereโs my line. Remember cuts the ๐ฆ-axis at five
yup. And itโs a negative slope, so
thatโs a downhill slope. Every time I increase my
๐ฅ-coordinate by one, the ๐ฆ-coordinate decreases by one; thatโs negative one ๐ฅ,
which matches my graph.

Now letโs plot ๐ฆ plus ๐ฅ equals
two. Well weโve a slight problem here
because this isnโt quite in this right format. Itโs not in our ๐ฆ equals something
times ๐ฅ plus another number format. So what Iโm gonna do is Iโm going
to subtract ๐ฅ from both sides of my equation, which gives me ๐ฆ plus ๐ฅ take away
๐ฅ on the left-hand side and two take away ๐ฅ on the right-hand side. So if I start off with positive ๐ฅ
and then I take away ๐ฅ, those two things are gonna cancel each other out. So Iโve just got ๐ฆ on the
left-hand side of my equation. Now on the right-hand side, Iโve
got two take away ๐ฅ; now it doesnโt matter whether I say two take away ๐ฅ or if I
start off with negative ๐ฅ and then I add two onto it. Remember this two on its own here
is really a positive two. And in this format, itโs really
easy to recognize that the negative ๐ฅ means the negative one ๐ฅ. So the slope is negative one. And that plus two on its own here
tells us that weโre cutting the ๐ฆ-axis at positive two. So I can put that on the graph. Here we go, cut the ๐ฆ-axis at
positive two and then every time I increase my ๐ฅ-coordinate by one, the ๐ฆ
coordinate decreases by one. So these are the points youโre
gonna generate. And when you join them up thatโs
what theyโre gonna look like.

Now itโs also worth noting at this
point Iโve been drawing all these little orange lines and showing you know increase
๐ฅ by one the ๐ฆ-coordinate goes down by one, you donโt actually need to be doing
those. So you only need to be plotting the
points. So Iโm just putting them on the
page so you can see them. It makes it nice and clear what
weโre doing โ how weโre counting this up, but you wouldnโt normally draw those
orange lines in when youโre 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 ๐ฆ equals ๐๐ฅ plus ๐ format, which makes it much easier to plot.

Letโs do another couple of examples
then.

So for this one, Iโve got ๐ฅ plus
๐ฆ plus three equals zero. Iโm gonna have to do a bit more
rearranging. So first of all, Iโm gonna subtract
๐ฅ from both sides. And when I do that, Iโve got ๐ฅ and
Iโm taking away ๐ฅ on the left-hand side, so I can cancel those two out. And on the right-hand side, I got
zero take away ๐ฅ. Well that is just negative ๐ฅ. So Iโve got ๐ฆ plus three is equal
to negative ๐ฅ, 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, Iโve got ๐ฆ add three take away three. So if I have to do three take away
three, thatโs nothing; so these two terms over here cancel out. So Iโve got ๐ฆ is equal to negative
๐ฅ take away three. Great, thatโs now in my ๐ฆ equals
๐๐ฅ plus ๐ format. And negative ๐ฅ is the same as
negative one ๐ฅ, so our slope is negative one. So cutting the ๐ฆ-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 ๐ฅ-coordinate
decreases our ๐ฆ-coordinate by one, leaves with these points which look like this
when I join them up. So thatโs the equation ๐ฅ plus ๐ฆ
plus three equals zero, which I rearranged to ๐ฆ equals negative ๐ฅ take away three
in order to quickly be able to see what the slope was, negative one, and where the
๐ฆ-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 ๐ฆ
equals ๐๐ฅ plus ๐ format so that we can easily work out the slope and the
intercept.

So to summarize then, ๐ฆ equals
๐๐ฅ plus ๐ or ๐๐ฅ plus ๐ is the general form of the straight line equation. The multiplier ๐ฅ is the slope,
which means that when I increase my ๐ฅ-coordinate by one, the ๐ฆ-coordinate will
increase by whatever that ๐ value is โ whether itโs positive or whether itโs
negative. And the ๐ value tells us the
๐ฆ-coordinate of the point on the line that cuts the ๐ฆ-axis. If the ๐ value was positive, weโll
have an uphill line that looks like this. And if it was negative, weโll have
a downhill line like that. Sometimes we need to rearrange the
equation to get the ๐ฆ equals ๐๐ฅ plus ๐ format and hence the slope and where it
cuts the ๐ฆ-axis. And lastly remember not all
equations rearrange into the ๐ฆ equals ๐๐ฅ plus ๐ format. Not all functions represent
straight line graphs.