Question

How do you find the power function through a given set of points?

Answer 1

Hint: We first explain the power function. We take its general formula. We use the given sets of points to find the relations between the unknowns. We solve them to find the specific formula. We take an example to understand the concept of power function better.

Complete step by step solution:
It is easy to confuse power functions with exponential functions. Both have a basic form that is given by two parameters. Both forms look very similar. In exponential functions, a fixed base is raised to a variable exponent. In power functions, however, a variable base is raised to a fixed exponent.
The general form of power function is $f\left( x \right)=a{{x}^{b}}$.
So, there are two unknowns for a power function.
We use the given sets of points in the form of $\left( x,y \right)$ where $y=f\left( x \right)=a{{x}^{b}}$.
We get the relations between the unknowns.
We solve them to find the power function.
The function for the area of a circle with radius $r$ is $A=\pi {{r}^{2}}$ which can be termed as a power function.
Equating with the function $f\left( x \right)=a{{x}^{b}}$, we get $a=\pi ,b=2$.

Note: It is worth to notice that if the value $b=0$ for $f\left( x \right)=a{{x}^{b}}$, then the function becomes constant where $f\left( x \right)=a$. We also can see that depending on the value of the power b being even and odd, the function becomes an even and odd function.