# Can you determine this function?

• Fjolvar
In summary, the conversation revolved around finding a function that could accurately fit a given red curve, with the ability to manipulate the y-intercept, maximum point value, and length of the curve. Different equations were suggested, including a 4-parameter fit and an even polynomial fit, with the final suggestion being a functional form involving an exponential term that would guarantee one maximum point and no minima or negative values.
Fjolvar
Help please - I can't determine this function.

Okay I will try to make this as straightforward and clear as possible. My calculus background is rusty because it's been awhile, so this may seem trivial.

I am trying to find a function that can best fit the shape of the red curve shown in the attached figure. I need to be able to choose the y-intercept and increase/decrease the maximum point value marked as "1" on both the y and x-axis location. I also need to be able to increase/decrease the length/x-value of the end portion of the curve marked as "2."

I have so far tried playing with quadratic functions, a sine wave (from 0 to π), and a double exponential. No luck so far. I would greatly appreciate any advice on this seemingly simple problem.

#### Attachments

• Function.png
1.2 KB · Views: 422
Last edited:
Fjolvar said:
Okay I will try to make this as straightforward and clear as possible. My calculus background is rusty because it's been awhile, so this may seem trivial.

I am trying to find a function that can best fit the shape of the red curve shown in the attached figure. I need to be able to choose the y-intercept and increase/decrease the maximum point value marked as "1" on both the y and x-axis location. I also need to be able to increase/decrease the length/x-value of the end portion of the curve marked as "2."

I have so far tried playing with quadratic functions, a sine wave (from 0 to π), and a double exponential. No luck so far. I would greatly appreciate any advice on this seemingly simple problem.

Maybe this 4 parameter fit will work:

$$y_0+a\frac{x}{L}\left(1-\frac{x}{L}\right)\left(1+c\frac{x}{L}\right)$$

Chestermiller said:
Maybe this 4 parameter fit will work:

$$y_0+a\frac{x}{L}\left(1-\frac{x}{L}\right)\left(1+c\frac{x}{L}\right)$$

Hmm I've been playing with this equation, but not having too much luck. Could you explain it? :)

Fjolvar said:
Hmm I've been playing with this equation, but not having too much luck. Could you explain it? :)

The first term is the value at x = 0. The parameter x=L is what you call point 2. Note that the function values at the x = 0 and x = L are both the same, as shown on your figure. The parameter a adjusts the height of the peak (along with the parameter c), and the parameter c moves the location of the peak around between x = 0 and x = L.

Maybe an even polynomial fit? Quartic or order 6? Course you may need a few more points to constrain it...

Chestermiller said:
Maybe this 4 parameter fit will work:

$$y_0+a\frac{x}{L}\left(1-\frac{x}{L}\right)\left(1+c\frac{x}{L}\right)$$

OK. I figured out an even better functional form than this previous one to try that I think you'll like much better:

$$y_0+a\frac{x}{L}\left(1-\frac{x}{L}\right)e^{c(\frac{x}{L})}$$

If c = 0, the maximum will be at x = L/2. If c < 0, the maximum will be at x < L/2, and if c > 0, the maximum will be at x > L/2. This functional form will guarantee one maximum and no minima (or negative values) between x = 0 and x = L.

Chet

## What is a function?

A function is a mathematical concept that describes the relationship between two variables, where the output value is determined by the input value. It can be represented by an equation or a graph.

## How do you determine a function?

To determine a function, you need to analyze the relationship between the input and output values. This can be done by looking at a table of values, a graph, or by using algebraic methods such as substitution and solving for the dependent variable.

## What is the difference between a linear and nonlinear function?

A linear function is a function where the graph forms a straight line, while a nonlinear function has a curved or non-linear graph. Linear functions have a constant rate of change, while nonlinear functions have a changing rate of change.

## How can you tell if a function is one-to-one?

A function is one-to-one if each input value has a unique output value and no two input values have the same output value. This can be determined by looking at a graph or by using the horizontal line test.

## What is the importance of determining a function?

Determining a function is important in many areas of science, such as physics, engineering, and economics. It allows us to model and predict real-world phenomena, make calculations, and solve problems. Understanding functions also helps in developing critical thinking and problem-solving skills.

Replies
3
Views
2K
Replies
6
Views
2K
Replies
2
Views
2K
Replies
5
Views
1K
Replies
5
Views
2K
Replies
8
Views
2K
Replies
26
Views
2K
Replies
4
Views
2K
Replies
3
Views
1K
Replies
3
Views
1K