# What is the equation of the function through these points?What are the equations of the functions passing through the following points (it's not linear)? (1, 24) (2, 192) (3,648) (4,1536)...

What is the equation of the function through these points?

What are the equations of the functions passing through the following points (it's not linear)? (1, 24) (2, 192) (3,648) (4,1536) (5,3000) And the one through the points (1,18) (2,72) (3,162) (4,288) (5,450) (6,648) (7,882) (8,1152) (9,1458) (10,1800) (11,2178) (12,2592) (13,3042) Thanks!

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### 1 Answer

These are both power functions; `y=ax^b` . You know that they are not linear (`(Deltay)/(Delta x)` is not constant).

One way to know the functions are power functions is to graph the points `(lnx,lny)` . If these points are on a straight line, the functions are exponential (`y=ab^x` ). Since they are not, you suspect that they are power functions.

To verify that they are indeed power functions, we can look at the finite differences. In the first case, the third order finite differences are constant indicating that the function is a cubic; in the second case the second order finite differences are constant indicating that the function is a quadratic.

x y d1 d2 d3

1 24

168

2 192 288

456 144

3 648 432

888 144

4 1536 576

1464

5 3000

**Using the regression capabilities of a graphing calculator, you find the function to be `y=24x^3` **

**The second order differences of the second list is 36; using the regression capabilities of a graphing calculator we find the equation to be `y=18x^2` **