Power models are very versatile. They can be used in many different cases. If your points draw a smile shape, have strong or gentle growth, or have a swing, the power model could be a good fit. Remember, each set of points has its own unique best model.
Because there’s an infinite amount of ways to create a collection of points that can be modeled as an expression with powers, there’s an infinite number of graphs that fit the expression below. The only difference between these graphs is the values of the coefficients $a$ and $n$.
Theory
Here are seven different graphs that match the expression above:
The values of $a$ and $n$ determine what the graph looks like. There are a bunch of different expressions that are all power functions. Just look at the figure above!
Below is a brief description of how the function will look for different values of $n$.
If $n$ is a positive even number, you get a parabola (pink graph).
If $n$ is a positive odd number, you get graphs that extend along the $y$ axis (purple graph).
If $n=0$, you get the straight line that intersects the $y$-axis at $y=a$.
If $n$ is a negative integer you get rational functions (yellow, orange and blue graphs).
If $n\in \mathbb{Q}$ ($n$ is a fraction) you get a root function (turquoise graph).
If $0<n<1$ you get a graph that starts at the origin and grows slowly outwards (green graph).