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MARKED 2009/06/07 linearrrr.JPG



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A LINEAR RELATION is a relation between two variables which creates a straight line when graphed on a cartesian plane or other coordinate system. LINEAR RELATIONS may have a positive (ex. 1) OR negative (ex. 2) slope. Also, the slope is of a linear relation is always constant. If the line of best fit is going in a positive direction, it is considered a positive incline. If the graph is going towards the top right corner of the graph, starting at the origin (0,0) ,the graph is linear (as shown below).
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A NON-LINEAR RELATION is a relation between two variables that does not create a straight line when graphed on a cartesian plain or coordinate system. In this case, the data is moving in a negative (downward) direction, so it has a negative decline. Non-linear relations do not have a constant slope. Linear relations usually occur when there's a relation between the X and Y values. However non-linear relations usually occur when there is little or no relation between the X and Y values.
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DIRECT VARIATION and PARTIAL VARIATION are examples of linear relation.
A Direct Variation is a relationship between two variables in which one variable is a constant multiple of the other (ex. y = 2x).
A Partial Variation is a relationship between two variables in which on variable is a constant multiple of the other, PLUS a fixed or constant term (ex. y = 2x + 5).

There are 3 main ways to determine whether a relationship is linear or non-linear:

  1. APPEARANCE - A linear relationship will always be a straight line, and a non-linear relationship will not be a straight line.
  2. EQUATION OF THE LINE - The equation of a linear line is always in the form of "y = mx + b", and has a degree of 1 (the degree is always the exponent of "x"). The equation of a non-linear line is not in the form of "y = mx + b", and has a degree of any number, other than one.
  3. FIRST DIFFERENCES - The first differences for a linear relationship are always equal. In a non-linear relationship, the first differences are not equal (see below on how to make a first differences chart).


The degree of an equation is always equal to the exponent of "x" in the equation of the line.

Equations


The Equation for a Linear Relation is in the form of Y = MX + B, and has to have a degree of one
(ex. y = 3x + 4)
If the exponent of x (degree) is not one, then the function is Non-Linear.
(ex. y = x3 - 1)
If the difference of the equations are all the same, then the relation is linear. If the difference is not the same, the relation is non-linear.

You can also put the equation in Standard Form. The equation for Standard Form should look like this:
Ax + By + C = 0.

The three rules are:
  • A must be positive
  • No fractions or decimals
  • Order is important

ex. Y=–3X+4
3X+Y=4
3X+Y–4=0


Sometimes you have to find the slope. The equation for that is m = Y2-Y1 / X2-X1
ex. m=8-2-1=6=2
4 3

Sometimes you have to find the equation of the line. The equation is Y-Y1=m(X - X1). Of course you need to know one point and the slope, so if you don't have that, then simply use the equation for slope first.
ex. y-4=3(X-2)
Y-4=3X-6
Y=3X-6+4
Y=3X-2


First Differences


The First Difference determines the difference between the X and Y axis. In order for this to happen the X axis coordinates have to be evenly spaced out.

When calculating First Differences and if the First Differences are the same the relation will be linear. If the First Differences are different it would be considered a non-linear relation.

How to Make a First Differences Chart

Steps:
1. Make a Chart to fit the appropriate headings; X, Y, and first differences.
2. Record all of the X and Y coordinates from the graph to the chart.
3. Subtract your first Y value from your second Y value and subtract the second Y value from the third etc.

Click to play video on how to make a first differences chart.
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Play Who Wants To Be A Millionaire...Linear v.s. Non-Linear Edition!





Citations


MathPower 9. Toronto: McGraw-Hill Ryerson, 1999.

Young, Tige. n.d. 17 Dec. 2008 <http://www.tigeyoung.com/stretching/bigpic.html>.

Rancourt, S. (2009). MPM 1D1 - Course Notes