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

A linear function is a function
of the form F of X equals

a X Plus B where A&B represent

real numbers. And when we show
this graphically, a represents

the gradients of the function
and B represents the Y axis

intersect, which is sometimes
called the vertical intercept.

Now what do you think would
happen if we varied a? Well,

let's have a look at a few

examples. Because we're looking
at the graphs of linear

functions, that means we're
going to be looking at straight

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

however, we often choose three
points because the Third Point

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

have a look at F of X equals X

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

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

S is one.

Is 1 + 2, which gives

us 3. An F of two.

2 + 2 which will give us

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

2 X +2.

F of X equals 2 X +2.

So we get F of 0.

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

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

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

which gives us 6.

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

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

X +2. We would have FO

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

F of one which gives us minus 2

* 1. Which is minus 2 +
2, which gives us 0.

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

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

Equals minus X

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

Equals 0 + 2, which is 2.

S is one which equals minus 1
+ 2, which equals 1. And finally

F of two which is minus 2
+ 2 which equals 0.

Now what we're interested in
doing is looking at the graphs

of these functions. So if we
have our axes drawn with F of X

on the vertical scale an X on
the horizontal axis, the first

function we looked at was F of X
equals X +2, which gave us

points at 02.

Second point resort.

13 Our third
points was at 2 full.

And when we join this up, we
expect a straight line.

We can

label less.
F of X.

Equals X +2.

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

Which games, the points 02,
which we've already marked here,

was the .1 four.

And it gave us the .2.

6.

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

We can label
SF of X.

Equals 2 X +2.

The next function we looked up
was F of X equals minus two X

+2, and once again this gave us
a points at 02 appoint at one

zero and a point at two and

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

expect a straight line.

We can label less.

F of X.

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

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

02 again point at one one.

Anna points AT20.

We can join those up to get
a straight line.

This is
F of

X. Equals minus
X plus so.

Now first thing we notice about
these graphs is that they all

crossed 2 on the F of X axis.
That's be'cause be value is 2 in

every single function and be
represents the Y axis intercept.

What we were interested in is
what happens as the value of a

changes. Now when A is positive,
the line goes up and the bigger

the value of A, the faster the
line goes up as X increases.

And when A is negative, the line

goes down. And the bigger the
value of an absolute terms, the

faster the line goes down as X

increases. OK, So what happens
as we very be?

Well, that's always good place
to start is by actually looking

at few examples. So let's
consider the example F of X

equals 2X plus 3.

F of 0 here would be 2
* 0 + 3, which is 0

+ 3, which is just three.

F of one is 2 * 1, which gives

Me 2. Plus three, which gives

me 5. An F of

two. Gives Me 2 * 2 which is 4.

Plus three, which gives me 7.

OK, Next One next functional
look at is F of X

equals 2X plus one.

OK, for this function we get F
of 0 is equal to 2 * 0, which

is 0 plus one, which gives me

one. I have one gives Me 2 * 1
which is 2 plus one which gives

me 3. And F of two gives
Me 2 * 2, which is 4 +

1, which gives me 5.

And the final function I want to
look at is F of X equals

2X minus three.

F of X equals 2X
minus three, so F of

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

3. F of one.

2 * 1 which is 2 takeaway. Three
gives me minus one and finally F

of two. Which is 2 * 2,
which is 4 takeaway three, which

gives me one.

So what we're interested in
doing is looking at the graphs

of these functions.

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

X one horizontal axis. So
first function we talked

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

we had were zero and three.

15

And two.

And Seven. We can join

those up. With a
straight line label

up F of
X equals 2X

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

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

Zero and one.

One and three.

Two and five.

Once again, we can draw those.

Join those up with a ruler.

Label at
one F

of X.
Equals 2X plus one.

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

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

One and minus one.

And two. And warm.

But enjoying those off.

As before.

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

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

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

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

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

when B was three.

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

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

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

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

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

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

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

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

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

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

We can sketch what they might

look like. F of X axis here.

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

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

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

Which comes through two on the F
of X axis.

So if of X equals 2.

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

That just comes through.

Minus three on F of X axis.

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

equals 0? But let's have a look.

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

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

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

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

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

These values of AF

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

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

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

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

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

because B equals 0.

And it has a gradients of minus

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

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

before, so it might be like

this. F of X
equals minus 3X.

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

form F of X equals a X plus B.

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

equals X plus B. It's quite
useful, so you can think about

now if we used Y equals F of X

just for convenience. So suppose

I had. 4X minus three

Y. Equals
2.

First thing we want to do is
make Y the subject of this

equation. So if I had three Y
answer both sides 4X equals 2 +

3 Y. Now I want to get three
wide by itself, so I need to

take away 2 from both sides.

So over here I got 4X
takeaway 2 on this side. If

I take away too, we just get
left with three Y.

And so finally to make why the
subject I need to divide both

sides by three.

So we get 4 thirds of X.

Minus 2/3 equals

Y. And as we said before,
Y equals F of X. So this means

our function is actually F of X
equals 4 thirds X minus 2/3.

So this function represents a
straight line with the gradients

of Four Thirds and Y axis
intercept of minus 2/3.

What about if we had two
X minus 8 Y plus eight

Y minus one equals 0?

Once again, we want to make why
the subject of the equation so a

natural first step would be to
add 1 to both sides.

So 2X plus eight Y
equals 1.

Next thing you want to do to get
8. Why by itself is to subtract

2 X from both sides. If we
subtract 2 actually miss side,

we just get left with a Y and
this side we get one takeaway

2X. And finally we need to
divide both sides by eight since

we just want why we've got

eight, why there? So divide
both sides by it. We got Y

equals 1/8 - 2 over 8 times X
and obviously ones are

functioning to form a X Plus B,
which means we would change

around. Just rearrange this
right son side here to get Y

equals minus 2 eighths of X
plus 1/8.

And we can simplify minus 2
eighths to be minus 1/4.

So we get minus one quarter of X
Plus one 8th. And as we said

before, Y equals F of X.

So here we have it.

We are function is F of X equals
minus one quarter X Plus one

8th, and graphically this is
represented by a straight line

with the gradients of minus 1/4
and yx intercept of 1/8.

What about if we
have this example?

Y equals. 13
X minus 8.

All divided by 5.

Now a little why is already the

subject of the formula. It's not
quite in the required form, and

that's because of this divide by

5. But we can just rewrite the
right hand side as Y equals 13 X

divided by 5 - 8 / 5 and
since why is F of X? We can

write this as F of X equals 13

over 5X. Minus 8 over

5. So this function is
represented graphically by a

straight line with the gradients
of 13 over 5 and a Y axis

intercept of minus 8 fifths.