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.