Return to Video

## www.mathcentre.ac.uk/.../Linear%20functions.mp4

• 0:02 - 0:09
A linear function is a function
of the form F of X equals
• 0:09 - 0:13
a X Plus B where A&B represent
• 0:13 - 0:18
real numbers. And when we show
this graphically, a represents
• 0:18 - 0:23
the gradients of the function
and B represents the Y axis
• 0:23 - 0:26
intersect, which is sometimes
called the vertical intercept.
• 0:27 - 0:31
Now what do you think would
happen if we varied a? Well,
• 0:31 - 0:34
let's have a look at a few
• 0:34 - 0:37
examples. Because we're looking
at the graphs of linear
• 0:37 - 0:41
functions, that means we're
going to be looking at straight
• 0:41 - 0:45
lines, and so plot a straight
line. We only need two points,
• 0:45 - 0:49
however, we often choose three
points because the Third Point
• 0:49 - 0:54
is a good check to make sure we
haven't made a mistake, so let's
• 0:54 - 0:58
have a look at F of X equals X
• 0:58 - 1:04
+2. OK, first points I look
at is F of 0.
• 1:04 - 1:10
Now F of zero 0 + 2,
which is simply too.
• 1:10 - 1:13
S is one.
• 1:13 - 1:17
Is 1 + 2, which gives
• 1:17 - 1:19
us 3. An F of two.
• 1:20 - 1:23
2 + 2 which will give us
• 1:23 - 1:31
4. OK, for the next function,
let's look at F of X equals
• 1:31 - 1:32
2 X +2.
• 1:32 - 1:36
F of X equals 2 X +2.
• 1:37 - 1:40
So we get F of 0.
• 1:40 - 1:45
Equals 2 * 0, which is 0 + 2,
which gives us 2.
• 1:46 - 1:54
S is one which gives us 2 *
1 which is 2 + 2, which gives
• 1:54 - 2:01
us 4. And F of two which gives
us 2 * 2, which is 4 + 2,
• 2:01 - 2:03
which gives us 6.
• 2:03 - 2:08
There's no reason why I
shouldn't be negative, so let's
• 2:08 - 2:15
look a few negative values. If
we had F of X equals minus two
• 2:15 - 2:19
X +2. We would have FO
• 2:19 - 2:26
equals. Minus 2 * 0 which is
0 + 2, which gives us 2.
• 2:26 - 2:29
F of one which gives us minus 2
• 2:29 - 2:35
* 1. Which is minus 2 +
2, which gives us 0.
• 2:35 - 2:43
An F of two which gives us minus
2 * 2 which is minus 4 +
• 2:43 - 2:49
2 which gives us minus two. And
finally we'll look at F of X.
• 2:49 - 2:52
Equals minus X
• 2:52 - 2:58
+2. So we've got
F of 0.
• 2:58 - 3:01
Equals 0 + 2, which is 2.
• 3:02 - 3:10
S is one which equals minus 1
+ 2, which equals 1. And finally
• 3:10 - 3:17
F of two which is minus 2
+ 2 which equals 0.
• 3:18 - 3:22
Now what we're interested in
doing is looking at the graphs
• 3:22 - 3:28
of these functions. So if we
have our axes drawn with F of X
• 3:28 - 3:33
on the vertical scale an X on
the horizontal axis, the first
• 3:33 - 3:39
function we looked at was F of X
equals X +2, which gave us
• 3:39 - 3:41
points at 02.
• 3:41 - 3:43
Second point resort.
• 3:44 - 3:50
13 Our third
points was at 2 full.
• 3:51 - 3:55
And when we join this up, we
expect a straight line.
• 3:56 - 3:59
We can
• 3:59 - 4:05
label less.
F of X.
• 4:05 - 4:09
Equals X +2.
• 4:10 - 4:15
The second function we looked up
was F of X equals 2 X +2.
• 4:16 - 4:21
Which games, the points 02,
which we've already marked here,
• 4:21 - 4:23
was the .1 four.
• 4:24 - 4:27
And it gave us the .2.
• 4:28 - 4:29
6.
• 4:30 - 4:35
We should be able to draw these
with a straight line.
• 4:37 - 4:44
We can label
SF of X.
• 4:45 - 4:48
Equals 2 X +2.
• 4:50 - 4:56
The next function we looked up
was F of X equals minus two X
• 4:56 - 5:02
+2, and once again this gave us
a points at 02 appoint at one
• 5:02 - 5:05
zero and a point at two and
• 5:05 - 5:10
minus 2. And when we join
these up as before, we
• 5:10 - 5:12
expect a straight line.
• 5:16 - 5:18
We can label less.
• 5:19 - 5:21
F of X.
• 5:21 - 5:28
Equals minus two X +2 and the
final function we looked at was
• 5:28 - 5:35
F of X equals minus X +2
and this gave us a point at
• 5:35 - 5:39
02 again point at one one.
• 5:40 - 5:44
Anna points AT20.
• 5:44 - 5:48
We can join those up to get
a straight line.
• 5:50 - 5:55
This is
F of
• 5:55 - 6:02
X. Equals minus
X plus so.
• 6:03 - 6:07
Now first thing we notice about
these graphs is that they all
• 6:07 - 6:13
crossed 2 on the F of X axis.
That's be'cause be value is 2 in
• 6:13 - 6:17
every single function and be
represents the Y axis intercept.
• 6:17 - 6:23
What we were interested in is
what happens as the value of a
• 6:23 - 6:29
changes. Now when A is positive,
the line goes up and the bigger
• 6:29 - 6:34
the value of A, the faster the
line goes up as X increases.
• 6:35 - 6:37
And when A is negative, the line
• 6:37 - 6:43
goes down. And the bigger the
value of an absolute terms, the
• 6:43 - 6:46
faster the line goes down as X
• 6:46 - 6:51
increases. OK, So what happens
as we very be?
• 6:52 - 6:58
Well, that's always good place
to start is by actually looking
• 6:58 - 7:04
at few examples. So let's
consider the example F of X
• 7:04 - 7:07
equals 2X plus 3.
• 7:07 - 7:15
F of 0 here would be 2
* 0 + 3, which is 0
• 7:15 - 7:18
+ 3, which is just three.
• 7:18 - 7:22
F of one is 2 * 1, which gives
• 7:22 - 7:26
Me 2. Plus three, which gives
• 7:26 - 7:29
me 5. An F of
• 7:29 - 7:33
two. Gives Me 2 * 2 which is 4.
• 7:34 - 7:37
Plus three, which gives me 7.
• 7:38 - 7:44
OK, Next One next functional
look at is F of X
• 7:44 - 7:46
equals 2X plus one.
• 7:48 - 7:55
OK, for this function we get F
of 0 is equal to 2 * 0, which
• 7:55 - 7:58
is 0 plus one, which gives me
• 7:58 - 8:05
one. I have one gives Me 2 * 1
which is 2 plus one which gives
• 8:05 - 8:12
me 3. And F of two gives
Me 2 * 2, which is 4 +
• 8:12 - 8:15
1, which gives me 5.
• 8:16 - 8:23
And the final function I want to
look at is F of X equals
• 8:23 - 8:24
2X minus three.
• 8:24 - 8:32
F of X equals 2X
minus three, so F of
• 8:32 - 8:38
0. Is 2 times here, which is
zero takeaway 3 which is minus
• 8:38 - 8:41
3. F of one.
• 8:41 - 8:49
2 * 1 which is 2 takeaway. Three
gives me minus one and finally F
• 8:49 - 8:55
of two. Which is 2 * 2,
which is 4 takeaway three, which
• 8:55 - 8:56
gives me one.
• 8:57 - 9:01
So what we're interested in
doing is looking at the graphs
• 9:01 - 9:02
of these functions.
• 9:02 - 9:07
So as usual, we have RF of
X on the vertical axis and
• 9:07 - 9:11
X one horizontal axis. So
first function we talked
• 9:11 - 9:16
about was F of X equals 2X
plus three and the points
• 9:16 - 9:18
we had were zero and three.
• 9:20 - 9:22
15
• 9:22 - 9:25
And two.
• 9:26 - 9:30
And Seven. We can join
• 9:30 - 9:37
those up. With a
straight line label
• 9:37 - 9:45
up F of
X equals 2X
• 9:45 - 9:51
plus 3. The next
function we looked up was F of X
• 9:51 - 9:54
equals 2X plus one and the
points we had there were.
• 9:55 - 9:57
Zero and one.
• 9:59 - 10:00
One and three.
• 10:01 - 10:04
Two and five.
• 10:06 - 10:09
Once again, we can draw those.
• 10:10 - 10:11
Join those up with a ruler.
• 10:12 - 10:17
Label at
one F
• 10:17 - 10:24
of X.
Equals 2X plus one.
• 10:25 - 10:30
And the final function looked up
was F of X equals 2X minus
• 10:30 - 10:35
three. And the points we had
were 0 - 3.
• 10:36 - 10:39
One and minus one.
• 10:40 - 10:42
And two. And warm.
• 10:43 - 10:46
But enjoying those off.
• 10:46 - 10:47
As before.
• 10:48 - 10:55
With a ruler. We label list we
get F of X equals 2X minus
• 10:55 - 11:00
three. OK, first thing we notice
here is that all the graphs are
• 11:00 - 11:05
parallel. In fact they have the
same gradients, and that's
• 11:05 - 11:11
because in each case the value
of a was two. So all the graphs
• 11:11 - 11:18
have a gradient of two and we
also notice that as we varied B,
• 11:18 - 11:20
when B was three.
• 11:20 - 11:23
The graph of the function went
through three on the F of X
• 11:23 - 11:28
axis. Would be was one the graph
of the function went through one
• 11:28 - 11:32
on the F of X axis and when be
was minus three. The graph of
• 11:32 - 11:36
the function went through minus
three on the F of X axis.
• 11:37 - 11:42
OK, so we know what happens when
I'm being positive and when A&B
• 11:42 - 11:46
are negative. What happens if
A&BRO? Well, let's see what
• 11:46 - 11:51
think about what happens when a
equals 0 first of all.
• 11:52 - 11:57
So if A equals 0 we get a
function of the form F of X
• 11:57 - 12:02
equals a constant, so that could
be for example, F of X equals 2.
• 12:03 - 12:08
Or F of X equals minus three.
Just a couple of examples.
• 12:09 - 12:12
We can sketch what they might
• 12:12 - 12:15
look like. F of X axis here.
• 12:15 - 12:20
Now X axis here F of X equals 2.
That means for.
• 12:21 - 12:24
Whatever the value of X,
the F of X values always
• 12:24 - 12:27
two. So in fact we just
get a horizontal line.
• 12:30 - 12:34
Which comes through two on the F
of X axis.
• 12:34 - 12:36
So if of X equals 2.
• 12:37 - 12:42
And when F of X equals minus
three, we get a horizontal line.
• 12:44 - 12:45
That just comes through.
• 12:46 - 12:49
Minus three on F of X axis.
• 12:49 - 12:56
So that's what happens when a
equals 0. What about when B
• 12:56 - 12:59
equals 0? But let's have a look.
• 13:01 - 13:06
The B equals 0. We get a
function of the form F of X
• 13:06 - 13:11
equals a X and as we said at the
beginning, a can be any real
• 13:11 - 13:18
number. So, for example,
we might have F of X
• 13:18 - 13:24
equals 2X or F of X
equals minus 3X.
• 13:25 - 13:28
OK, and as we've already said,
what happens when we use?
• 13:29 - 13:32
These values of AF
• 13:32 - 13:38
of X&X. For looking at F of
X equals 2X. It's going to come
• 13:38 - 13:43
through the origin because B
equals 0, so it will cross F of
• 13:43 - 13:49
X at 0. And it will have a
gradient of two since a IS2.
• 13:50 - 13:56
So it's a sketch. This could
represent F of X equals 2X.
• 13:57 - 14:01
Of X equals minus three X once
again will go through the origin
• 14:01 - 14:03
because B equals 0.
• 14:04 - 14:05
And it has a gradients of minus
• 14:05 - 14:09
three. Remember the minus means
the line is coming down and the
• 14:09 - 14:12
three means that it's going to
be a bit steeper than it was
• 14:12 - 14:14
before, so it might be like
• 14:14 - 14:21
this. F of X
equals minus 3X.
• 14:22 - 14:26
OK, Lastly I want to look at
functions which are not in the
• 14:26 - 14:30
form F of X equals a X plus B.
• 14:30 - 14:36
So. What would we do? So we want
our functions in form F of X
• 14:36 - 14:40
equals X plus B. It's quite
useful, so you can think about
• 14:40 - 14:43
now if we used Y equals F of X
• 14:43 - 14:46
just for convenience. So suppose
• 14:46 - 14:50
I had. 4X minus three
• 14:50 - 14:54
Y. Equals
2.
• 14:55 - 15:01
First thing we want to do is
make Y the subject of this
• 15:01 - 15:07
equation. So if I had three Y
answer both sides 4X equals 2 +
• 15:07 - 15:14
3 Y. Now I want to get three
wide by itself, so I need to
• 15:14 - 15:16
take away 2 from both sides.
• 15:16 - 15:21
So over here I got 4X
takeaway 2 on this side. If
• 15:21 - 15:25
I take away too, we just get
left with three Y.
• 15:26 - 15:31
And so finally to make why the
subject I need to divide both
• 15:31 - 15:32
sides by three.
• 15:33 - 15:35
So we get 4 thirds of X.
• 15:36 - 15:39
Minus 2/3 equals
• 15:39 - 15:46
Y. And as we said before,
Y equals F of X. So this means
• 15:46 - 15:52
our function is actually F of X
equals 4 thirds X minus 2/3.
• 15:53 - 15:58
So this function represents a
straight line with the gradients
• 15:58 - 16:04
of Four Thirds and Y axis
intercept of minus 2/3.
• 16:06 - 16:14
What about if we had two
X minus 8 Y plus eight
• 16:14 - 16:17
Y minus one equals 0?
• 16:18 - 16:23
Once again, we want to make why
the subject of the equation so a
• 16:23 - 16:27
natural first step would be to
add 1 to both sides.
• 16:27 - 16:33
So 2X plus eight Y
equals 1.
• 16:35 - 16:40
Next thing you want to do to get
8. Why by itself is to subtract
• 16:40 - 16:45
2 X from both sides. If we
subtract 2 actually miss side,
• 16:45 - 16:50
we just get left with a Y and
this side we get one takeaway
• 16:50 - 16:55
2X. And finally we need to
divide both sides by eight since
• 16:55 - 16:56
we just want why we've got
• 16:56 - 17:02
eight, why there? So divide
both sides by it. We got Y
• 17:02 - 17:08
equals 1/8 - 2 over 8 times X
and obviously ones are
• 17:08 - 17:14
functioning to form a X Plus B,
which means we would change
• 17:14 - 17:19
around. Just rearrange this
right son side here to get Y
• 17:19 - 17:23
equals minus 2 eighths of X
plus 1/8.
• 17:24 - 17:29
And we can simplify minus 2
eighths to be minus 1/4.
• 17:29 - 17:36
So we get minus one quarter of X
Plus one 8th. And as we said
• 17:36 - 17:39
before, Y equals F of X.
• 17:40 - 17:41
So here we have it.
• 17:41 - 17:47
We are function is F of X equals
minus one quarter X Plus one
• 17:47 - 17:51
8th, and graphically this is
represented by a straight line
• 17:51 - 17:56
with the gradients of minus 1/4
and yx intercept of 1/8.
• 17:56 - 18:01
What about if we
have this example?
• 18:03 - 18:09
Y equals. 13
X minus 8.
• 18:09 - 18:11
All divided by 5.
• 18:12 - 18:14
Now a little why is already the
• 18:14 - 18:18
subject of the formula. It's not
quite in the required form, and
• 18:18 - 18:19
that's because of this divide by
• 18:19 - 18:27
5. But we can just rewrite the
right hand side as Y equals 13 X
• 18:27 - 18:34
divided by 5 - 8 / 5 and
since why is F of X? We can
• 18:34 - 18:38
write this as F of X equals 13
• 18:38 - 18:41
over 5X. Minus 8 over
• 18:41 - 18:46
5. So this function is
represented graphically by a
• 18:46 - 18:51
straight line with the gradients
of 13 over 5 and a Y axis
• 18:51 - 18:54
intercept of minus 8 fifths.
Title:
www.mathcentre.ac.uk/.../Linear%20functions.mp4
Video Language:
English

• Uploaded
mathcentre