Recall
Linear relations can be graphed as a single straight line. In a table of values, a linear relation has constant first differences.
First differences are the differences between consecutive \(y\)-values in a table of values. This is often denoted as \(y_2-y_1\) or \(\Delta y\). First differences can only be used to identify a linear relation if the \(x\)-values in the table are increasing or decreasing by constant values.
Observe the following graph and table. Both of these represent the same linear relation.
Notice that the \(x\)-values are increasing by a constant value of \(2\).
1) You can expalin what you mean by your last point to the learner in reference to the table. Perhaps using the word 'pattern' as you have previously -MC
If we plot the Group Size, \(x\), and the Number of Handshakes, \(y\).
The fact that all of the second differences are the same means that the relation between the number of handshakes and the group size is what is known as a quadratic relation.
A quadratic relation is a non-linear relation, with constant second differences. Its graph forms a curve called a parabola.
This graph shows a quadratic relation; the curve is known as a parabola.
The following table represents a quadratic relation. Calculate the second difference for this relation.
The second difference is .
In a group of \(10\) people, everyone is going to shake everyone else's hand exactly once. How many handshakes will take place?
One way to solve this problem is to extend the table relating group size and number of handshakes (seen earlier) until it reaches a group size of \(10\). Since the second differences are always \(1\), the first differences will increase by \(1\) each time. This will allow us to determine the next \(y\)-values in the table.
From the table, we can see that a group of \(10\) individuals would need \(45\) handshakes.
Note that this is only one of many effective methods of solving this problem!