Stephanie M. answered • 07/16/15
Tutor
5.0
(914)
Degree in Math with 5+ Years of Tutoring Experience
Exponential functions increase faster than either linear or quadratic functions, while square root functions increase slower than either linear or quadratic functions. We can use a handy trick to find out whether this data set matches a linear or quadratic function. Look at the differences between successive y values:
0
> 1
1
> 2
3
> 3
6
> 4
10
> 5
15
Since the differences are not the same, this function is not linear. Now we look at the differences between the differences...
0
> 1
1 > 1
> 2
3 > 1
> 3
6 > 1
> 4
10 > 1
> 5
15
> 1
1 > 1
> 2
3 > 1
> 3
6 > 1
> 4
10 > 1
> 5
15
Since these differences are the same, this is a quadratic function. If it took one more iteration, the function would be cubic. One more would be quartic (x4), then quintic (x5), etc.
If the function were growing too quickly for any of those, it would be exponential (given our choices). Too slowly for even a linear function would be a square root function (given our choices).
Quartic (x4):
0
> 1
1 > 14
> 15 > 36
16 > 50 > 24
> 65 > 60
81 > 110 > 24
> 175 > 84
256 > 194
> 369
625
It takes four iterations for the differences to settle down.
Exponential (2x):
1
> 1
2 > 1
> 2 > 1
4 > 2
> 4 > 2
8 > 4
> 8 > 4
16 > 8
> 16
32
Notice that this pattern will never settle down to the same differences.
Square Root (√x):
0
> 1
1
> 0.414...
√2
> 0.318...
√3
> 0.268...
2
> 0.236...
√5
This pattern will never settle down either, especially since the differences will all eventually become irrational numbers. Unlike the exponential function, though, the differences become smaller as x gets larger.
