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Power function

A power function is a function of the form:

f(x)=b*x^a

Its domain is the set of non-negative real numbers \mathbbR_+.
Its codomain is also the set of non-negative real numbers \mathbbR_+.

The relationship between the swinging time and the length of a pendulum is, for instance, given by a power function.

How does the constant a affect a power function?

0 < a < 1: The power function is growing with decreasing slope.

a > 1: The power function is growing with increasing slope.

a < 0: The power function is decaying.

a = 1: It’s a linear function.

a = 2: It’s a quadratic function.

If two points (x1,y1) and (x2,y2) are given, you can calculate the constant a with the formula:

a=frac{log(frac{y_2}{y_1})}{log(frac{x_2}{x_1})}=frac{log(y_2)-log(y_1)}{log(x_2)-log(x_1)}

How does the constant b affect a power function?

The graph of a power function is passing through the point (1,b).

If the constant a and one of the graph’s points (x1,y1) is given, it is possible to calculate the constant b:

b=frac{y_1}{{x_1^a}}=frac{y_2}{{x_2^a}}

The multiply-multiply property of power functions

Given a power function of the form f(x)=b*x^a. According to the multiply-multiply property you will have to multiply the function by ka, if you multiply x by the constant k:

f(k*x)=k^a*f(x)

For example: Given the function:

f(x)=3*x^2

If the functions input is x=2, the outcome is 12:

f(x)=3*2^2=12

Now we multiply x by 3, whereby we get the outcome 108:

f(x)=3*(2*3)^2=108

Had we multiplied the result, 12, we got in the first place, by ka= 32=9 instead, we would have achieved the same result:

9*12=108

Calculate the constants a and b of a power function with two points given

Please enter the coordinates of two points.

Point 1
x1: y1:
Point 2
x2: y2:

Recognizing a power trend in a data set

If you have two sets of data, and you want to know if there is a power trend in the relationship between them, you can plot them within a coordinate system, both axes of which have logarithmic scales. The more the points fit to a straight line, the closer to a power trend they are.