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00:00:01,860 --> 00:00:09,270
A linear function is a function
of the form F of X equals

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00:00:09,270 --> 00:00:13,260
a X Plus B where A&B represent

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00:00:13,260 --> 00:00:18,236
real numbers. And when we show
this graphically, a represents

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the gradients of the function
and B represents the Y axis

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intersect, which is sometimes
called the vertical intercept.

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00:00:26,730 --> 00:00:31,218
Now what do you think would
happen if we varied a? Well,

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00:00:31,218 --> 00:00:33,836
let's have a look at a few

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00:00:33,836 --> 00:00:37,202
examples. Because we're looking
at the graphs of linear

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00:00:37,202 --> 00:00:40,942
functions, that means we're
going to be looking at straight

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lines, and so plot a straight
line. We only need two points,

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however, we often choose three
points because the Third Point

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is a good check to make sure we
haven't made a mistake, so let's

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have a look at F of X equals X

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+2. OK, first points I look
at is F of 0.

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Now F of zero 0 + 2,
which is simply too.

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S is one.

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Is 1 + 2, which gives

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us 3. An F of two.

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2 + 2 which will give us

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4. OK, for the next function,
let's look at F of X equals

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2 X +2.

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F of X equals 2 X +2.

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So we get F of 0.

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Equals 2 * 0, which is 0 + 2,
which gives us 2.

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S is one which gives us 2 *
1 which is 2 + 2, which gives

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us 4. And F of two which gives
us 2 * 2, which is 4 + 2,

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which gives us 6.

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There's no reason why I
shouldn't be negative, so let's

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look a few negative values. If
we had F of X equals minus two

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X +2. We would have FO

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equals. Minus 2 * 0 which is
0 + 2, which gives us 2.

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00:02:25,700 --> 00:02:29,156
F of one which gives us minus 2

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00:02:29,156 --> 00:02:35,470
* 1. Which is minus 2 +
2, which gives us 0.

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An F of two which gives us minus
2 * 2 which is minus 4 +

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2 which gives us minus two. And
finally we'll look at F of X.

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Equals minus X

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+2. So we've got
F of 0.

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00:02:57,580 --> 00:03:01,318
Equals 0 + 2, which is 2.

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00:03:02,170 --> 00:03:10,136
S is one which equals minus 1
+ 2, which equals 1. And finally

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00:03:10,136 --> 00:03:16,964
F of two which is minus 2
+ 2 which equals 0.

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00:03:17,510 --> 00:03:22,207
Now what we're interested in
doing is looking at the graphs

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00:03:22,207 --> 00:03:28,185
of these functions. So if we
have our axes drawn with F of X

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00:03:28,185 --> 00:03:33,309
on the vertical scale an X on
the horizontal axis, the first

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00:03:33,309 --> 00:03:39,287
function we looked at was F of X
equals X +2, which gave us

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00:03:39,287 --> 00:03:40,568
points at 02.

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00:03:41,320 --> 00:03:42,859
Second point resort.

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00:03:43,600 --> 00:03:50,357
13 Our third
points was at 2 full.

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00:03:50,960 --> 00:03:55,426
And when we join this up, we
expect a straight line.

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00:03:56,170 --> 00:03:59,414
We can

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label less.
F of X.

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00:04:05,420 --> 00:04:09,188
Equals X +2.

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The second function we looked up
was F of X equals 2 X +2.

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00:04:15,970 --> 00:04:20,890
Which games, the points 02,
which we've already marked here,

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00:04:20,890 --> 00:04:22,858
was the .1 four.

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00:04:23,560 --> 00:04:26,998
And it gave us the .2.

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00:04:27,890 --> 00:04:28,930
6.

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We should be able to draw these
with a straight line.

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We can label
SF of X.

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00:04:44,540 --> 00:04:48,460
Equals 2 X +2.

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00:04:49,510 --> 00:04:55,810
The next function we looked up
was F of X equals minus two X

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00:04:55,810 --> 00:05:02,110
+2, and once again this gave us
a points at 02 appoint at one

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zero and a point at two and

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minus 2. And when we join
these up as before, we

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expect a straight line.

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We can label less.

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F of X.

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Equals minus two X +2 and the
final function we looked at was

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F of X equals minus X +2
and this gave us a point at

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00:05:35,387 --> 00:05:38,513
02 again point at one one.

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00:05:39,950 --> 00:05:43,739
Anna points AT20.

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00:05:44,350 --> 00:05:48,360
We can join those up to get
a straight line.

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This is
F of

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00:05:55,410 --> 00:06:01,650
X. Equals minus
X plus so.

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00:06:02,580 --> 00:06:07,380
Now first thing we notice about
these graphs is that they all

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00:06:07,380 --> 00:06:13,380
crossed 2 on the F of X axis.
That's be'cause be value is 2 in

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00:06:13,380 --> 00:06:17,380
every single function and be
represents the Y axis intercept.

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00:06:17,380 --> 00:06:22,580
What we were interested in is
what happens as the value of a

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00:06:22,580 --> 00:06:28,556
changes. Now when A is positive,
the line goes up and the bigger

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00:06:28,556 --> 00:06:33,730
the value of A, the faster the
line goes up as X increases.

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00:06:34,580 --> 00:06:37,359
And when A is negative, the line

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00:06:37,359 --> 00:06:43,080
goes down. And the bigger the
value of an absolute terms, the

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faster the line goes down as X

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00:06:46,146 --> 00:06:51,060
increases. OK, So what happens
as we very be?

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00:06:51,900 --> 00:06:58,137
Well, that's always good place
to start is by actually looking

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00:06:58,137 --> 00:07:04,374
at few examples. So let's
consider the example F of X

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00:07:04,374 --> 00:07:06,642
equals 2X plus 3.

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00:07:07,290 --> 00:07:15,004
F of 0 here would be 2
* 0 + 3, which is 0

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00:07:15,004 --> 00:07:18,310
+ 3, which is just three.

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00:07:18,390 --> 00:07:21,972
F of one is 2 * 1, which gives

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00:07:21,972 --> 00:07:25,792
Me 2. Plus three, which gives

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00:07:25,792 --> 00:07:29,236
me 5. An F of

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00:07:29,236 --> 00:07:33,194
two. Gives Me 2 * 2 which is 4.

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00:07:33,810 --> 00:07:37,020
Plus three, which gives me 7.

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00:07:37,850 --> 00:07:44,142
OK, Next One next functional
look at is F of X

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00:07:44,142 --> 00:07:46,430
equals 2X plus one.

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00:07:47,670 --> 00:07:54,662
OK, for this function we get F
of 0 is equal to 2 * 0, which

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00:07:54,662 --> 00:07:57,721
is 0 plus one, which gives me

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00:07:57,721 --> 00:08:04,640
one. I have one gives Me 2 * 1
which is 2 plus one which gives

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00:08:04,640 --> 00:08:12,387
me 3. And F of two gives
Me 2 * 2, which is 4 +

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00:08:12,387 --> 00:08:14,832
1, which gives me 5.

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00:08:15,530 --> 00:08:22,642
And the final function I want to
look at is F of X equals

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00:08:22,642 --> 00:08:24,166
2X minus three.

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00:08:24,180 --> 00:08:31,790
F of X equals 2X
minus three, so F of

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0. Is 2 times here, which is
zero takeaway 3 which is minus

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00:08:37,960 --> 00:08:40,580
3. F of one.

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00:08:41,170 --> 00:08:48,535
2 * 1 which is 2 takeaway. Three
gives me minus one and finally F

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00:08:48,535 --> 00:08:55,087
of two. Which is 2 * 2,
which is 4 takeaway three, which

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gives me one.

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00:08:56,980 --> 00:09:01,083
So what we're interested in
doing is looking at the graphs

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00:09:01,083 --> 00:09:02,202
of these functions.

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So as usual, we have RF of
X on the vertical axis and

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X one horizontal axis. So
first function we talked

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about was F of X equals 2X
plus three and the points

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we had were zero and three.

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15

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And two.

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And Seven. We can join

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00:09:29,876 --> 00:09:37,475
those up. With a
straight line label

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up F of
X equals 2X

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plus 3. The next
function we looked up was F of X

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equals 2X plus one and the
points we had there were.

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Zero and one.

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One and three.

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Two and five.

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Once again, we can draw those.

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Join those up with a ruler.

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Label at
one F

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of X.
Equals 2X plus one.

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And the final function looked up
was F of X equals 2X minus

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three. And the points we had
were 0 - 3.

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One and minus one.

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And two. And warm.

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But enjoying those off.

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As before.

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With a ruler. We label list we
get F of X equals 2X minus

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three. OK, first thing we notice
here is that all the graphs are

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parallel. In fact they have the
same gradients, and that's

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because in each case the value
of a was two. So all the graphs

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have a gradient of two and we
also notice that as we varied B,

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when B was three.

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The graph of the function went
through three on the F of X

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axis. Would be was one the graph
of the function went through one

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on the F of X axis and when be
was minus three. The graph of

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the function went through minus
three on the F of X axis.

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OK, so we know what happens when
I'm being positive and when A&B

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are negative. What happens if
A&BRO? Well, let's see what

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think about what happens when a
equals 0 first of all.

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So if A equals 0 we get a
function of the form F of X

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equals a constant, so that could
be for example, F of X equals 2.

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Or F of X equals minus three.
Just a couple of examples.

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We can sketch what they might

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look like. F of X axis here.

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Now X axis here F of X equals 2.
That means for.

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Whatever the value of X,
the F of X values always

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two. So in fact we just
get a horizontal line.

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Which comes through two on the F
of X axis.

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So if of X equals 2.

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And when F of X equals minus
three, we get a horizontal line.

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That just comes through.

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Minus three on F of X axis.

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So that's what happens when a
equals 0. What about when B

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00:12:55,734 --> 00:12:59,423
equals 0? But let's have a look.

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The B equals 0. We get a
function of the form F of X

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equals a X and as we said at the
beginning, a can be any real

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number. So, for example,
we might have F of X

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equals 2X or F of X
equals minus 3X.

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OK, and as we've already said,
what happens when we use?

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These values of AF

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of X&X. For looking at F of
X equals 2X. It's going to come

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through the origin because B
equals 0, so it will cross F of

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X at 0. And it will have a
gradient of two since a IS2.

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So it's a sketch. This could
represent F of X equals 2X.

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Of X equals minus three X once
again will go through the origin

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because B equals 0.

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And it has a gradients of minus

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three. Remember the minus means
the line is coming down and the

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three means that it's going to
be a bit steeper than it was

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before, so it might be like

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this. F of X
equals minus 3X.

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OK, Lastly I want to look at
functions which are not in the

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00:14:26,497 --> 00:14:29,638
form F of X equals a X plus B.

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So. What would we do? So we want
our functions in form F of X

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00:14:36,348 --> 00:14:40,212
equals X plus B. It's quite
useful, so you can think about

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00:14:40,212 --> 00:14:43,110
now if we used Y equals F of X

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00:14:43,110 --> 00:14:45,970
just for convenience. So suppose

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00:14:45,970 --> 00:14:49,930
I had. 4X minus three

187
00:14:49,930 --> 00:14:53,990
Y. Equals
2.

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00:14:55,060 --> 00:15:01,014
First thing we want to do is
make Y the subject of this

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00:15:01,014 --> 00:15:07,426
equation. So if I had three Y
answer both sides 4X equals 2 +

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00:15:07,426 --> 00:15:13,549
3 Y. Now I want to get three
wide by itself, so I need to

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00:15:13,549 --> 00:15:15,607
take away 2 from both sides.

192
00:15:16,470 --> 00:15:20,922
So over here I got 4X
takeaway 2 on this side. If

193
00:15:20,922 --> 00:15:25,003
I take away too, we just get
left with three Y.

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00:15:26,360 --> 00:15:31,053
And so finally to make why the
subject I need to divide both

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00:15:31,053 --> 00:15:32,136
sides by three.

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00:15:32,750 --> 00:15:35,228
So we get 4 thirds of X.

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00:15:35,740 --> 00:15:38,770
Minus 2/3 equals

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00:15:38,770 --> 00:15:46,271
Y. And as we said before,
Y equals F of X. So this means

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00:15:46,271 --> 00:15:52,212
our function is actually F of X
equals 4 thirds X minus 2/3.

200
00:15:53,100 --> 00:15:58,490
So this function represents a
straight line with the gradients

201
00:15:58,490 --> 00:16:03,880
of Four Thirds and Y axis
intercept of minus 2/3.

202
00:16:05,770 --> 00:16:13,582
What about if we had two
X minus 8 Y plus eight

203
00:16:13,582 --> 00:16:16,837
Y minus one equals 0?

204
00:16:17,660 --> 00:16:22,616
Once again, we want to make why
the subject of the equation so a

205
00:16:22,616 --> 00:16:26,510
natural first step would be to
add 1 to both sides.

206
00:16:27,100 --> 00:16:32,630
So 2X plus eight Y
equals 1.

207
00:16:34,680 --> 00:16:40,215
Next thing you want to do to get
8. Why by itself is to subtract

208
00:16:40,215 --> 00:16:44,643
2 X from both sides. If we
subtract 2 actually miss side,

209
00:16:44,643 --> 00:16:49,809
we just get left with a Y and
this side we get one takeaway

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00:16:49,809 --> 00:16:54,517
2X. And finally we need to
divide both sides by eight since

211
00:16:54,517 --> 00:16:56,419
we just want why we've got

212
00:16:56,419 --> 00:17:02,236
eight, why there? So divide
both sides by it. We got Y

213
00:17:02,236 --> 00:17:07,924
equals 1/8 - 2 over 8 times X
and obviously ones are

214
00:17:07,924 --> 00:17:13,612
functioning to form a X Plus B,
which means we would change

215
00:17:13,612 --> 00:17:18,826
around. Just rearrange this
right son side here to get Y

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00:17:18,826 --> 00:17:22,618
equals minus 2 eighths of X
plus 1/8.

217
00:17:23,710 --> 00:17:28,704
And we can simplify minus 2
eighths to be minus 1/4.

218
00:17:29,230 --> 00:17:36,490
So we get minus one quarter of X
Plus one 8th. And as we said

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00:17:36,490 --> 00:17:39,394
before, Y equals F of X.

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00:17:39,530 --> 00:17:40,550
So here we have it.

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00:17:41,100 --> 00:17:46,938
We are function is F of X equals
minus one quarter X Plus one

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00:17:46,938 --> 00:17:51,108
8th, and graphically this is
represented by a straight line

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00:17:51,108 --> 00:17:55,695
with the gradients of minus 1/4
and yx intercept of 1/8.

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00:17:56,240 --> 00:18:01,448
What about if we
have this example?

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00:18:02,600 --> 00:18:08,768
Y equals. 13
X minus 8.

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00:18:09,280 --> 00:18:11,060
All divided by 5.

227
00:18:12,160 --> 00:18:13,840
Now a little why is already the

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00:18:13,840 --> 00:18:17,648
subject of the formula. It's not
quite in the required form, and

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00:18:17,648 --> 00:18:19,394
that's because of this divide by

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00:18:19,394 --> 00:18:26,798
5. But we can just rewrite the
right hand side as Y equals 13 X

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00:18:26,798 --> 00:18:33,950
divided by 5 - 8 / 5 and
since why is F of X? We can

232
00:18:33,950 --> 00:18:37,526
write this as F of X equals 13

233
00:18:37,526 --> 00:18:41,296
over 5X. Minus 8 over

234
00:18:41,296 --> 00:18:46,186
5. So this function is
represented graphically by a

235
00:18:46,186 --> 00:18:51,477
straight line with the gradients
of 13 over 5 and a Y axis

236
00:18:51,477 --> 00:18:53,512
intercept of minus 8 fifths.