0:00:01.860,0:00:09.270
A linear function is a function[br]of the form F of X equals

0:00:09.270,0:00:13.260
a X Plus B where A&B represent

0:00:13.260,0:00:18.236
real numbers. And when we show[br]this graphically, a represents

0:00:18.236,0:00:22.823
the gradients of the function[br]and B represents the Y axis

0:00:22.823,0:00:26.159
intersect, which is sometimes[br]called the vertical intercept.

0:00:26.730,0:00:31.218
Now what do you think would[br]happen if we varied a? Well,

0:00:31.218,0:00:33.836
let's have a look at a few

0:00:33.836,0:00:37.202
examples. Because we're looking[br]at the graphs of linear

0:00:37.202,0:00:40.942
functions, that means we're[br]going to be looking at straight

0:00:40.942,0:00:45.430
lines, and so plot a straight[br]line. We only need two points,

0:00:45.430,0:00:49.170
however, we often choose three[br]points because the Third Point

0:00:49.170,0:00:54.406
is a good check to make sure we[br]haven't made a mistake, so let's

0:00:54.406,0:00:57.772
have a look at F of X equals X

0:00:57.772,0:01:04.170
+2. OK, first points I look[br]at is F of 0.

0:01:04.200,0:01:10.404
Now F of zero 0 + 2,[br]which is simply too.

0:01:10.490,0:01:13.118
S is one.

0:01:13.120,0:01:16.612
Is 1 + 2, which gives

0:01:16.612,0:01:19.180
us 3. An F of two.

0:01:19.690,0:01:23.246
2 + 2 which will give us

0:01:23.246,0:01:30.726
4. OK, for the next function,[br]let's look at F of X equals

0:01:30.726,0:01:32.430
2 X +2.

0:01:32.440,0:01:36.388
F of X equals 2 X +2.

0:01:36.950,0:01:40.328
So we get F of 0.

0:01:40.330,0:01:45.166
Equals 2 * 0, which is 0 + 2,[br]which gives us 2.

0:01:46.010,0:01:53.706
S is one which gives us 2 *[br]1 which is 2 + 2, which gives

0:01:53.706,0:02:01.130
us 4. And F of two which gives[br]us 2 * 2, which is 4 + 2,

0:02:01.130,0:02:02.762
which gives us 6.

0:02:03.330,0:02:08.040
There's no reason why I[br]shouldn't be negative, so let's

0:02:08.040,0:02:14.634
look a few negative values. If[br]we had F of X equals minus two

0:02:14.634,0:02:18.580
X +2. We would have FO

0:02:18.580,0:02:25.641
equals. Minus 2 * 0 which is[br]0 + 2, which gives us 2.

0:02:25.700,0:02:29.156
F of one which gives us minus 2

0:02:29.156,0:02:35.470
* 1. Which is minus 2 +[br]2, which gives us 0.

0:02:35.470,0:02:42.638
An F of two which gives us minus[br]2 * 2 which is minus 4 +

0:02:42.638,0:02:48.910
2 which gives us minus two. And[br]finally we'll look at F of X.

0:02:48.930,0:02:52.086
Equals minus X

0:02:52.086,0:02:57.580
+2. So we've got[br]F of 0.

0:02:57.580,0:03:01.318
Equals 0 + 2, which is 2.

0:03:02.170,0:03:10.136
S is one which equals minus 1[br]+ 2, which equals 1. And finally

0:03:10.136,0:03:16.964
F of two which is minus 2[br]+ 2 which equals 0.

0:03:17.510,0:03:22.207
Now what we're interested in[br]doing is looking at the graphs

0:03:22.207,0:03:28.185
of these functions. So if we[br]have our axes drawn with F of X

0:03:28.185,0:03:33.309
on the vertical scale an X on[br]the horizontal axis, the first

0:03:33.309,0:03:39.287
function we looked at was F of X[br]equals X +2, which gave us

0:03:39.287,0:03:40.568
points at 02.

0:03:41.320,0:03:42.859
Second point resort.

0:03:43.600,0:03:50.357
13 Our third[br]points was at 2 full.

0:03:50.960,0:03:55.426
And when we join this up, we[br]expect a straight line.

0:03:56.170,0:03:59.414
We can

0:03:59.414,0:04:04.688
label less.[br]F of X.

0:04:05.420,0:04:09.188
Equals X +2.

0:04:09.930,0:04:15.376
The second function we looked up[br]was F of X equals 2 X +2.

0:04:15.970,0:04:20.890
Which games, the points 02,[br]which we've already marked here,

0:04:20.890,0:04:22.858
was the .1 four.

0:04:23.560,0:04:26.998
And it gave us the .2.

0:04:27.890,0:04:28.930
6.

0:04:30.230,0:04:35.213
We should be able to draw these[br]with a straight line.

0:04:37.240,0:04:44.470
We can label[br]SF of X.

0:04:44.540,0:04:48.460
Equals 2 X +2.

0:04:49.510,0:04:55.810
The next function we looked up[br]was F of X equals minus two X

0:04:55.810,0:05:02.110
+2, and once again this gave us[br]a points at 02 appoint at one

0:05:02.110,0:05:05.260
zero and a point at two and

0:05:05.260,0:05:10.387
minus 2. And when we join[br]these up as before, we

0:05:10.387,0:05:11.959
expect a straight line.

0:05:15.880,0:05:18.228
We can label less.

0:05:18.730,0:05:21.319
F of X.

0:05:21.320,0:05:28.093
Equals minus two X +2 and the[br]final function we looked at was

0:05:28.093,0:05:35.387
F of X equals minus X +2[br]and this gave us a point at

0:05:35.387,0:05:38.513
02 again point at one one.

0:05:39.950,0:05:43.739
Anna points AT20.

0:05:44.350,0:05:48.360
We can join those up to get[br]a straight line.

0:05:49.610,0:05:55.410
This is[br]F of

0:05:55.410,0:06:01.650
X. Equals minus[br]X plus so.

0:06:02.580,0:06:07.380
Now first thing we notice about[br]these graphs is that they all

0:06:07.380,0:06:13.380
crossed 2 on the F of X axis.[br]That's be'cause be value is 2 in

0:06:13.380,0:06:17.380
every single function and be[br]represents the Y axis intercept.

0:06:17.380,0:06:22.580
What we were interested in is[br]what happens as the value of a

0:06:22.580,0:06:28.556
changes. Now when A is positive,[br]the line goes up and the bigger

0:06:28.556,0:06:33.730
the value of A, the faster the[br]line goes up as X increases.

0:06:34.580,0:06:37.359
And when A is negative, the line

0:06:37.359,0:06:43.080
goes down. And the bigger the[br]value of an absolute terms, the

0:06:43.080,0:06:46.146
faster the line goes down as X

0:06:46.146,0:06:51.060
increases. OK, So what happens[br]as we very be?

0:06:51.900,0:06:58.137
Well, that's always good place[br]to start is by actually looking

0:06:58.137,0:07:04.374
at few examples. So let's[br]consider the example F of X

0:07:04.374,0:07:06.642
equals 2X plus 3.

0:07:07.290,0:07:15.004
F of 0 here would be 2[br]* 0 + 3, which is 0

0:07:15.004,0:07:18.310
+ 3, which is just three.

0:07:18.390,0:07:21.972
F of one is 2 * 1, which gives

0:07:21.972,0:07:25.792
Me 2. Plus three, which gives

0:07:25.792,0:07:29.236
me 5. An F of

0:07:29.236,0:07:33.194
two. Gives Me 2 * 2 which is 4.

0:07:33.810,0:07:37.020
Plus three, which gives me 7.

0:07:37.850,0:07:44.142
OK, Next One next functional[br]look at is F of X

0:07:44.142,0:07:46.430
equals 2X plus one.

0:07:47.670,0:07:54.662
OK, for this function we get F[br]of 0 is equal to 2 * 0, which

0:07:54.662,0:07:57.721
is 0 plus one, which gives me

0:07:57.721,0:08:04.640
one. I have one gives Me 2 * 1[br]which is 2 plus one which gives

0:08:04.640,0:08:12.387
me 3. And F of two gives[br]Me 2 * 2, which is 4 +

0:08:12.387,0:08:14.832
1, which gives me 5.

0:08:15.530,0:08:22.642
And the final function I want to[br]look at is F of X equals

0:08:22.642,0:08:24.166
2X minus three.

0:08:24.180,0:08:31.790
F of X equals 2X[br]minus three, so F of

0:08:31.790,0:08:37.960
0. Is 2 times here, which is[br]zero takeaway 3 which is minus

0:08:37.960,0:08:40.580
3. F of one.

0:08:41.170,0:08:48.535
2 * 1 which is 2 takeaway. Three[br]gives me minus one and finally F

0:08:48.535,0:08:55.087
of two. Which is 2 * 2,[br]which is 4 takeaway three, which

0:08:55.087,0:08:56.398
gives me one.

0:08:56.980,0:09:01.083
So what we're interested in[br]doing is looking at the graphs

0:09:01.083,0:09:02.202
of these functions.

0:09:02.210,0:09:07.488
So as usual, we have RF of[br]X on the vertical axis and

0:09:07.488,0:09:11.142
X one horizontal axis. So[br]first function we talked

0:09:11.142,0:09:16.014
about was F of X equals 2X[br]plus three and the points

0:09:16.014,0:09:18.450
we had were zero and three.

0:09:19.640,0:09:21.700
15

0:09:22.300,0:09:25.200
And two.

0:09:25.930,0:09:29.876
And Seven. We can join

0:09:29.876,0:09:37.475
those up. With a[br]straight line label

0:09:37.475,0:09:44.909
up F of[br]X equals 2X

0:09:44.909,0:09:51.140
plus 3. The next[br]function we looked up was F of X

0:09:51.140,0:09:54.429
equals 2X plus one and the[br]points we had there were.

0:09:54.940,0:09:56.710
Zero and one.

0:09:58.650,0:10:00.309
One and three.

0:10:00.910,0:10:04.420
Two and five.

0:10:06.370,0:10:08.836
Once again, we can draw those.

0:10:09.790,0:10:11.236
Join those up with a ruler.

0:10:12.040,0:10:17.432
Label at[br]one F

0:10:17.432,0:10:24.390
of X.[br]Equals 2X plus one.

0:10:25.290,0:10:30.451
And the final function looked up[br]was F of X equals 2X minus

0:10:30.451,0:10:34.818
three. And the points we had[br]were 0 - 3.

0:10:36.200,0:10:38.780
One and minus one.

0:10:39.760,0:10:42.060
And two. And warm.

0:10:42.560,0:10:45.628
But enjoying those off.

0:10:46.210,0:10:47.150
As before.

0:10:48.200,0:10:54.564
With a ruler. We label list we[br]get F of X equals 2X minus

0:10:54.564,0:11:00.466
three. OK, first thing we notice[br]here is that all the graphs are

0:11:00.466,0:11:05.006
parallel. In fact they have the[br]same gradients, and that's

0:11:05.006,0:11:11.362
because in each case the value[br]of a was two. So all the graphs

0:11:11.362,0:11:17.718
have a gradient of two and we[br]also notice that as we varied B,

0:11:17.718,0:11:19.534
when B was three.

0:11:20.090,0:11:23.171
The graph of the function went[br]through three on the F of X

0:11:23.171,0:11:27.622
axis. Would be was one the graph[br]of the function went through one

0:11:27.622,0:11:32.062
on the F of X axis and when be[br]was minus three. The graph of

0:11:32.062,0:11:35.614
the function went through minus[br]three on the F of X axis.

0:11:36.830,0:11:42.199
OK, so we know what happens when[br]I'm being positive and when A&B

0:11:42.199,0:11:46.329
are negative. What happens if[br]A&BRO? Well, let's see what

0:11:46.329,0:11:50.872
think about what happens when a[br]equals 0 first of all.

0:11:51.570,0:11:57.135
So if A equals 0 we get a[br]function of the form F of X

0:11:57.135,0:12:02.329
equals a constant, so that could[br]be for example, F of X equals 2.

0:12:03.360,0:12:08.208
Or F of X equals minus three.[br]Just a couple of examples.

0:12:08.770,0:12:12.118
We can sketch what they might

0:12:12.118,0:12:14.560
look like. F of X axis here.

0:12:15.170,0:12:20.378
Now X axis here F of X equals 2.[br]That means for.

0:12:20.930,0:12:24.120
Whatever the value of X,[br]the F of X values always

0:12:24.120,0:12:27.020
two. So in fact we just[br]get a horizontal line.

0:12:29.950,0:12:33.850
Which comes through two on the F[br]of X axis.

0:12:34.400,0:12:36.158
So if of X equals 2.

0:12:36.930,0:12:42.065
And when F of X equals minus[br]three, we get a horizontal line.

0:12:44.040,0:12:45.320
That just comes through.

0:12:46.000,0:12:48.877
Minus three on F of X axis.

0:12:49.410,0:12:55.734
So that's what happens when a[br]equals 0. What about when B

0:12:55.734,0:12:59.423
equals 0? But let's have a look.

0:13:00.670,0:13:05.766
The B equals 0. We get a[br]function of the form F of X

0:13:05.766,0:13:11.226
equals a X and as we said at the[br]beginning, a can be any real

0:13:11.226,0:13:18.155
number. So, for example,[br]we might have F of X

0:13:18.155,0:13:23.960
equals 2X or F of X[br]equals minus 3X.

0:13:25.160,0:13:28.262
OK, and as we've already said,[br]what happens when we use?

0:13:29.020,0:13:32.232
These values of AF

0:13:32.232,0:13:38.464
of X&X. For looking at F of[br]X equals 2X. It's going to come

0:13:38.464,0:13:42.650
through the origin because B[br]equals 0, so it will cross F of

0:13:42.650,0:13:48.510
X at 0. And it will have a[br]gradient of two since a IS2.

0:13:50.300,0:13:55.748
So it's a sketch. This could[br]represent F of X equals 2X.

0:13:56.520,0:14:01.434
Of X equals minus three X once[br]again will go through the origin

0:14:01.434,0:14:02.946
because B equals 0.

0:14:03.610,0:14:05.325
And it has a gradients of minus

0:14:05.325,0:14:09.173
three. Remember the minus means[br]the line is coming down and the

0:14:09.173,0:14:12.462
three means that it's going to[br]be a bit steeper than it was

0:14:12.462,0:14:13.980
before, so it might be like

0:14:13.980,0:14:20.510
this. F of X[br]equals minus 3X.

0:14:21.960,0:14:26.497
OK, Lastly I want to look at[br]functions which are not in the

0:14:26.497,0:14:29.638
form F of X equals a X plus B.

0:14:30.150,0:14:36.348
So. What would we do? So we want[br]our functions in form F of X

0:14:36.348,0:14:40.212
equals X plus B. It's quite[br]useful, so you can think about

0:14:40.212,0:14:43.110
now if we used Y equals F of X

0:14:43.110,0:14:45.970
just for convenience. So suppose

0:14:45.970,0:14:49.930
I had. 4X minus three

0:14:49.930,0:14:53.990
Y. Equals[br]2.

0:14:55.060,0:15:01.014
First thing we want to do is[br]make Y the subject of this

0:15:01.014,0:15:07.426
equation. So if I had three Y[br]answer both sides 4X equals 2 +

0:15:07.426,0:15:13.549
3 Y. Now I want to get three[br]wide by itself, so I need to

0:15:13.549,0:15:15.607
take away 2 from both sides.

0:15:16.470,0:15:20.922
So over here I got 4X[br]takeaway 2 on this side. If

0:15:20.922,0:15:25.003
I take away too, we just get[br]left with three Y.

0:15:26.360,0:15:31.053
And so finally to make why the[br]subject I need to divide both

0:15:31.053,0:15:32.136
sides by three.

0:15:32.750,0:15:35.228
So we get 4 thirds of X.

0:15:35.740,0:15:38.770
Minus 2/3 equals

0:15:38.770,0:15:46.271
Y. And as we said before,[br]Y equals F of X. So this means

0:15:46.271,0:15:52.212
our function is actually F of X[br]equals 4 thirds X minus 2/3.

0:15:53.100,0:15:58.490
So this function represents a[br]straight line with the gradients

0:15:58.490,0:16:03.880
of Four Thirds and Y axis[br]intercept of minus 2/3.

0:16:05.770,0:16:13.582
What about if we had two[br]X minus 8 Y plus eight

0:16:13.582,0:16:16.837
Y minus one equals 0?

0:16:17.660,0:16:22.616
Once again, we want to make why[br]the subject of the equation so a

0:16:22.616,0:16:26.510
natural first step would be to[br]add 1 to both sides.

0:16:27.100,0:16:32.630
So 2X plus eight Y[br]equals 1.

0:16:34.680,0:16:40.215
Next thing you want to do to get[br]8. Why by itself is to subtract

0:16:40.215,0:16:44.643
2 X from both sides. If we[br]subtract 2 actually miss side,

0:16:44.643,0:16:49.809
we just get left with a Y and[br]this side we get one takeaway

0:16:49.809,0:16:54.517
2X. And finally we need to[br]divide both sides by eight since

0:16:54.517,0:16:56.419
we just want why we've got

0:16:56.419,0:17:02.236
eight, why there? So divide[br]both sides by it. We got Y

0:17:02.236,0:17:07.924
equals 1/8 - 2 over 8 times X[br]and obviously ones are

0:17:07.924,0:17:13.612
functioning to form a X Plus B,[br]which means we would change

0:17:13.612,0:17:18.826
around. Just rearrange this[br]right son side here to get Y

0:17:18.826,0:17:22.618
equals minus 2 eighths of X[br]plus 1/8.

0:17:23.710,0:17:28.704
And we can simplify minus 2[br]eighths to be minus 1/4.

0:17:29.230,0:17:36.490
So we get minus one quarter of X[br]Plus one 8th. And as we said

0:17:36.490,0:17:39.394
before, Y equals F of X.

0:17:39.530,0:17:40.550
So here we have it.

0:17:41.100,0:17:46.938
We are function is F of X equals[br]minus one quarter X Plus one

0:17:46.938,0:17:51.108
8th, and graphically this is[br]represented by a straight line

0:17:51.108,0:17:55.695
with the gradients of minus 1/4[br]and yx intercept of 1/8.

0:17:56.240,0:18:01.448
What about if we[br]have this example?

0:18:02.600,0:18:08.768
Y equals. 13[br]X minus 8.

0:18:09.280,0:18:11.060
All divided by 5.

0:18:12.160,0:18:13.840
Now a little why is already the

0:18:13.840,0:18:17.648
subject of the formula. It's not[br]quite in the required form, and

0:18:17.648,0:18:19.394
that's because of this divide by

0:18:19.394,0:18:26.798
5. But we can just rewrite the[br]right hand side as Y equals 13 X

0:18:26.798,0:18:33.950
divided by 5 - 8 / 5 and[br]since why is F of X? We can

0:18:33.950,0:18:37.526
write this as F of X equals 13

0:18:37.526,0:18:41.296
over 5X. Minus 8 over

0:18:41.296,0:18:46.186
5. So this function is[br]represented graphically by a

0:18:46.186,0:18:51.477
straight line with the gradients[br]of 13 over 5 and a Y axis

0:18:51.477,0:18:53.512
intercept of minus 8 fifths.