Linear Functions

Linear functions.

Alright! now that we built the basis (defining functions and linear equations) we can talk about linear functions, don't mix that up with linear transformations, we will talk about that eventually. Now it will come to light why the word linear. At this point, it would be good if you get some graphing software, such as Matlab, Derive, Mathematica, Octave or even google. Some of them are free, some not, Matlab can get quite expensive depending on the license you buy. I will make a post about some of them, eventually, and for this one, I will use Derive, because it is the simplest one. We will see about that later.

How does a linear function look like? well, it is a function, so it has to relate two sets, and satisfy the condition that, for every element of the domain, there is one and only one corresponding element in the codomain.

y = mx +b  <-- this is the "face" of a linear function. Why m? why b? what is variable? what is fixed?

m and b are fixed numbers, what I wrote is the slope intercept form, we'll see about that name.

For now, m and b are fixed numbers, x and y are the variables. So, an example of a linear function is 

y=3x+2 or y=7.5x+8/4.

This is not the only face a linear function can have. You can have 

Ax+By = C <--- this is called the standard form

you can go from one to the other performing the same operations on both sides as we did in the linear equations post. Let's see how to go from the standard to the slope intercept

Ax+By = C  <---- we need to get y alone on the left side, so let's subtract Ax from both sides

By = -Ax +C  <--- now let's divide both sides by B

y = -A/Bx +C/B

so, m = -A/B and b = C/B.

Alright, that is how they can look like. They are equations, because there is an equal sign somewhere in there, and they are linear, because no variable is raised to no power (remember there is an implicit hidden power of 1, x^1= x ) and no variable is input of any weird function, such as cos(x) or e^x or Ln(x) or such. So, linear.

Ok, what we can do with these things? can we solve them? the answer is  no, we can't.

There are infinite pairs (x,y) that satisfy  y = x +4, think about it.

If x is 1, y is 5, so (1,5) is a possible solution, but if x is 2, y is 6, so (2,6) might as well be the solution. So hopefully you understand that it doesn't make any sense to say "solve y = x+4"

So, what can we do with these linear functions? graph them, solve for particular values of x OR y.

And that's pretty much it, so let's graph this. How can we do that?

Well, as we saw, there are infinte pairs that make that equation true, if we set a two dimesional space, where x is the horizontal coordinate and y is the vertical coordinate, the pair (1,5) determines a point in that plane, and so does (2,6) and (3,7) and they don't have to be integers, (1.5,5.5) also satisfies the equation. So let's graph all those points along with a couple more (that I get just by selecting random x's and seeing what y ends up being) and see what we get

.

This is something we can actually do with pen and paper, and hopefully you will see that if you join the points you get a straight line, now I'm not going to prove that, but I can show you a graph that I get form Derive if I input y=x+4

What Derive does is basically plot thousands of points, now we don't got time for that, and we are smart. If we have two points, we can draw a line, right? that is all we need, two points, a pen, paper and a ruler, more than two points is redundant, hopefully that makes sense.

And this is the reason that they are called linear, because the graph is a line.

Ok, there are a few more elements to discuss, and the reason why the slope intercept form is "better" than the standard (it is easier to do some things with the standard though)

First, what gives the name to the slope intercept form. Remember it was  y=mx+b

m is the slope, we will talk about that later

and b "is the intercept" Acutally, (0,b) is the y intercept, if you notice, at the y axis, x is 0 (because x is the shift with respect to the y-axis, positive to the right, negative to the left)

So, just by looking at it, we can tell the slope and we can tell the y-intercept (point of intersection of the line and the y axis).

The slope is the number with the x, and it makes sense to call it slope too.

Picture this scenario, we have  y = mx, no b (or b=0, however you want to see it)

there is a fixed point here,  (0,0) no matter what m is, if x is 0, y is 0.

So, say m is 1, y=x, we can get the point (1,1) and draw a line with that and (0,0)

but if m is 2? we get (1,2) instead, and if m=100, we get (1,100) let's see how that looks like.

the green is y=x, the purlpe y=2x and the brownish is y=100x

so, as the slope grows, the line gets steeper, and that is what we think about when we see the word slope, right? a big slope, a very steep thing, so if the slope is 100, it should be steep, compared to what? well, to a slope of 1, if the slope is small, say  0.0001 , then we should expect almost no slope,

if the slope is 0, we should have an horizontal line (that is what one imangines with "no slope" )

and if the slope is negative, the line "goes down" ( as x increases y decreases)

so a slope of -100 for example, would be very steep, but pointing down.

Hopefully all this makes sense, I hope to get some feedback. I will put a couple of problems, but we will come back to this, in a more practical sense. Let me know what you think in the comments.

1)  3x +2y  = 7   <---- what is the slope here?

2)  5y = 3+x <--- write this in the slope intercept form

3) 5x+7y = 11 <--- what is the y-intercept here?