# How to modify a power function?

• SHRock
In summary, to find the values of a and b in the function y = ax^b, you can substitute the given ordered pairs into the equation to get two equations in the two unknowns. Then, by dividing one equation by the other, you can solve for b. Once you have b, you can use either equation to find a. If working with multiple ordered pairs, you can use nonlinear regression to get a more accurate estimate of the values for a and b.
SHRock

## Homework Statement

Let's say I have 2 ordered pairs. I want to find the the a and b in the function. How would I do that?

y=ax^b

## The Attempt at a Solution

y/a=x^b
b=log(y/a)/logx

But now I have 2 variables. How would I find those variables?

SHRock said:

## Homework Statement

Let's say I have 2 ordered pairs. I want to find the the a and b in the function. How would I do that?

y=ax^b

## The Attempt at a Solution

y/a=x^b
b=log(y/a)/logx

But now I have 2 variables. How would I find those variables?

Substitute the points you have into your equation. Then you will have two equations in the two unknowns, a and b.

Mark44 said:
Substitute the points you have into your equation. Then you will have two equations in the two unknowns, a and b.

Yeah but the problem is I don't know how to reduce that.

Let's say I pick 2 points

(35,0.01)
(4.5,30)

so the functions will be

b=log(0.01/a)/log 35

b=log(30/a)log4.5

Let's go back to the original equation, y = axb

.01 = a * 35b
30 = a* 4.5b

Dividing the first equation by the second, we have
.01/30 = 35b/4.5b = (35/4.5)b

Can you solve that for b? Once you get b, then use either equation to find a.

Mark44 said:
Let's go back to the original equation, y = axb

.01 = a * 35b
30 = a* 4.5b

Dividing the first equation by the second, we have
.01/30 = 35b/4.5b = (35/4.5)b

Can you solve that for b? Once you get b, then use either equation to find a.

Oh ok thanks I get it. Now another little questions. Let's say if I do this for multiple ordered pairs. And a and b aren't always the same, but pretty close. Should get the average of those and plug it in as b? Because this is like a lab and error will occur.

## 1. What is a power function?

A power function is a mathematical function of the form f(x) = ax^b, where a and b are constants and x is the independent variable. It is a type of exponential function that is commonly used in mathematics and science to model relationships between variables.

## 2. How can I modify a power function?

There are several ways to modify a power function, depending on what you want to achieve. Some common modifications include changing the values of a and b, adding or subtracting constants from the function, and applying different operations such as multiplication or division to the function.

## 3. What is the purpose of modifying a power function?

The purpose of modifying a power function is to alter its shape or behavior to better fit a given set of data or to model a specific relationship between variables. Modifying a power function can also help in simplifying calculations or making predictions based on the function.

## 4. Are there any limitations to modifying a power function?

Yes, there are some limitations to modifying a power function. For example, changing the value of b can alter the nature of the function, making it non-monotonic or non-differentiable. Also, modifying a power function too much may result in a poor fit to the data or an unrealistic relationship between variables.

## 5. Can I modify a power function to fit any type of data?

While a power function can be modified to fit a wide range of data, it may not always be the best choice for every data set. Other types of functions, such as logarithmic or polynomial functions, may be more suitable for certain types of data. It is important to carefully analyze the data and consider different function options before modifying a power function.

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