Lesson BASICS of linear functions

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1.1 Introduction



Any linear equation y=mx + c, where m and c are just constants (ie numbers) describes a straight line.

When a straight line equation is written in this form, the m and c values mean something on the graph:


  • m = the gradient (slope)

  • c = the y-intercept (where the line crosses or cuts the y-axis).



so,
y=3x+2 means: the line has a gradient of +3 and
crosses the y-axis at y=2.

and
y=-2x-7 means: the line has a gradient of -2 and
crosses the y-axis at y=-7.


EXAMPLE:
1. Find the gradient and y-intercept of the equation 12x-3y=15

solution:
First, re-write this in the form y=mx+c, so we get... 12x-15 = 3y or, the other way round, 3y = 12x-15.

Now divide by 3, to give y=4x-5.

So, the gradient is +4 and the y-intercept is -5.

1.2 Gradient


What does the gradient look like? What is the definition of the gradient?


Well, definition first. For 2 points (x1, y1) and (x2, y2)

Gradient+=+change_in_y%2Fchange_in_x+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

This definition applies to any straight line or curve or physical entity. Always.

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....

As gradient, m is defined as (y2-y1)/(x2-x1)

we get either:
m+=+%286-2%29%2F%283-1%29+=+4%2F2+=+2+
or
m+=+%282-6%29%2F%281-3%29+=+%28-4%29%2F%28-2%29+=+2+

Looking at these 2 ways of getting the same answer, one of them is simpler than the other. If you pick the easy way then good. If not, do not worry... stick to the definition and do not make a mistake with your negative signs and you will get the right answer.

1.3 Visualising Gradients



A positive gradient is any line like /
A negative gradient is any line like \
Example of y = 2x+3, with positive slope: +graph%28+100%2C+80%2C+-5%2C+5%2C+-10%2C+10%2C+2x%2B3%29+
Example of y = -x+2, with negative slope: +graph%28+100%2C+80%2C+-5%2C+5%2C+-5%2C+5%2C+-x%2B2%29+
A gradient of 1 is a line drawn at 45 degrees.
A gradient of 2 is a steeper line than this (but not 90degrees!).
A gradient of 3 is steeper still... etc.
A gradient of 1/2 is a line less than 45 degrees (but not as small as 22.5 degrees).
On this graph, y=x is red, y=2x is green, y=3x is blue, y=4x is violet: graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%2C+2x%2C+3x%2C+4x%29+
As you can see, the angle is not a linear function of gradient... gradient 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 gradient of 1. So be aware of that.

If you want to know, for curiosity's sake:
The angle for gradient=2 is about 63 degrees.
The angle for gradient=3 is about 72 degrees.

When you sketch, you are doing a SKETCH! so do not get a protractor out to get the correct angle, just draw a valid / or a \ line and then mark on the points where the line cuts the axes.
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