I forgot a title! Silly me.

Hello all of you!  You may address me Professor Hiles, or Stewart if you feel like it.  I’ll be taking a little bit to cover Linear Functions today.  These functions are incredibly simple, and there’s no doubt you’ve noticed them in your daily life before.  (Hint:  straight lines)  But describing linear functions in math terms requires a little bit more detail.

                Linear functions are functions that have a constant, repeating slope, resulting in graphs that show straight lines over an indefinite period.  These functions are applied to explain variables that result in a constant rate of change, for example.

                Now, any linear equation y=mx + c, where m and c are just constants (ie numbers) describes a straight line.  The “m” represents the gradient of the graph, or the ‘slope’.  Higher m values result in steeper slopes, and lower m values result in flatter slopes.  The “c” value represent where the graph crosses the Y-axis.

For example, if you have the function y = 8x + 4,

--You have a relatively steep Slope of 8 (if you’re range is limited)
--The graph crosses the Y-axis at y = 4.

Now, if you have a function with a negative slope, such as y = -2x -3

--You have a negative, downwards slope of 2

--your graph crosses the Y-axis at y = -3

A positive slope is any line like /
A negative slope is any line like \

A slope of 1 is a line drawn at 45 degrees.
A slope of 2 is a steeper line than this (but not 90degrees!).
A slope of 3 is steeper still... etc.
A slope of 1/2 is a line less than 45 degrees (but not as small as 22.5 degrees)

But sometimes, the world is cruel, and we aren’t provided a slope. 

But don’t fret!  If you are provided at least two points, you can find the gradient (basically another word for slope) between the two points using the function

f(x) = ^{change in y} / _{change in x} = ^{y2 – y1} / _{x2 – x1}

EXAMPLE:
1. Given the 2 points (1,2) and (3,6), what is the gradient between these 2 points?

Solution:
The order of the 2 points does not matter!  You pick one pair and take the other pair of coordinates away. Or you do it the opposite way. It does not matter. Just decide yourself and stick with it.

I shall show you both, to prove that the order of the points does not matter...

we get either:

or

To wrap up:

On this graph, y=x is red, y=2x is green, y=3x is blue, y=4x is violet:
As you can see, the angle is not a linear function of slope... the slope of 2 is twice as steep but that does not mean the angle the line makes with the floor is doubled when you go from a slope of 1. So be aware of that. 

Now you should do the problems in your book I've assigned!