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Calculating the first and second differences of a non-linear relation.
Let's look at an example.
Below is a table of values for the following non-linear equation.
y equals x squared plus 1.
Calculate the first differences for the data by subtracting the y values.
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So, 2 minus 5 equals
minus 3 and 1 minus
2 equals minus 1.
And 2 minus 1, equals 1 and 5
minus 2, equals 3.
Next calculate, the second differences for
the data by subtracting the first difference.
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Negative 1 minus a minus 3.
Negative 1 minus a minus 3 equals
negative 1 plus 3, which equals 2
and then 1 minus a minus 1.
1 minus, minus 1= 1+1 which equals 2.
And 3 minus 1.
3 minus 1.
Equals 2.
Since the first differences are not constant and the second
differences are constant, the graph for this equation is non-linear.
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Let's plot the points on the graph.
This is the x axis and this is the y axis.
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So let's plot our points for x and y.
Negative 2 and up 5 on the y axis.
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Then negative 1 and 2,
zero and 1, 1 and 2.
2 and 5.
Draw a line connecting these points.
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The shape of this graph is non-linear and is called a parabola.
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Three reasons why the graph is non-linear for
the equation y equals x squared plus 1.
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The first reason is, I graphed the points and the graph is not a straight line.
So, graph
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is not a straight line.
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The second reason is, in this example the exponent on the x is 2.
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So, x is raised
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to an exponent other than 1.
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And the third reason is, first differences are not constant.
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