Quadratics
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The Quadratic Relationship
Throughout this unit, there is one major type of relationship that we need to be aware of. It is known as the "quadratic relationship". In our previous years of mathematical studies, we have learnt about both the linear and non-linear relationships. From our past knowledge that we had acquired in maths class, we understand that linear relationships have EQUAL FIRST DIFFERENCES. Furthermore, we also understand that non-linear relationships DO NOT HAVE EQUAL FIRST DIFFERENCES. However, the "quadratic relationship" is something that we do not yet know. Therefore, it is in this section, in which we will be examining this relationship to a further degree.
QUADRATIC RELATIONSHIP
If we closely observe the table of values above, we should be able to notice a column titled "second differences". The "second differences" are something we have not yet heard of, but are simply the differences between the "first differences". They are crucial in determining whether or not a relationship can or cannot be classified as "quadratic". The table of values above shows a "quadratic relationship". Therefore, from looking at the table of values we can conclude that a "quadratic relationship" has EQUAL SECOND DIFFERENCES. In order to prove this statement, the data in the table of values above has been graphed, and the resulting graph has been shown below. Please take a look below the graph for the explanation.
From the previous section ("Exploring Parabolas") we learnt that the parabola is a common element throughout the quadratics unit. Therefore, it would be logical to assume that if a relation was to be "quadratic", the resulting graph would be in the shape of a parabola. If we take a look at the graph above, we can see that it is clearly in the shape of a parabola. Therefore, if a relationship were to be classifed as "quadratic" it should have equal second differences (table of values), and when the values are graphed, the resulting shape should be that of a parabola.