Expotential function

[chickeneps]

I'm looking for an equation that allows values of x to go from 0.0 to 1.0 in a "harder", linear, and "easier" fashion.

For example, I know that linear means y = x. But I'd like to know how to write harder and easier equations. This is an example of a harder equation, expressed in pairs:

(0,0) (.1,.05) (.2,.1.1) (.3,.23) (.4,.32) (.5,.41) (.6,.5) (.7,.61) (.8,.71) (.9,.93) (1,1)

This on a graph probably looks horrible, and I was just guessing. But I'd like a equation form where the equation could make it harder to get to 1.0 (that is, a steeper slope, where .5=.4 or something lower), or easier to get to 1.0 (that is, a shallower slope, where .5=.6 or something higher).

 

[HallsofIvy]

I'm looking for an equation that allows values of x to go from 0.0 to 1.0 in a "harder", linear, and "easier" fashion.

For example, I know that linear means y = x.

No, that is NOT what linear means. A "linear equation" is of the form y= ax+ b for numbers a and b.

But I'd like to know how to write harder and easier equations. This is an example of a harder equation, expressed in pairs:

(0,0) (.1,.05) (.2,.1.1) (.3,.23) (.4,.32) (.5,.41) (.6,.5) (.7,.61) (.8,.71) (.9,.93) (1,1)

?? That's not an equation at all so I have no idea what you mean by "an example of a harder equation".

This on a graph probably looks horrible, and I was just guessing. But I'd like a equation form where the equation could make it harder to get to 1.0 (that is, a steeper slope, where .5=.4 or something lower), or easier to get to 1.0 (that is, a shallower slope, where .5=.6 or something higher).

Nor do I know what you intended by ".5= .4" or ".5= .6"! If by "harder" and "easier" equations you simply mean "lines with steeper or lower slopes", just use y= ax+ b with a larger or smaller, respectively.

I have feeling you mean something more than this but I don't know what. Could you rephrase this? What do you mean by "harder equation" and "easier equation"? And I just realized you titled this "exponential function". So you mean something like $\displaystyle y= ab^x$? But the "slope" of such an equation varies from point to point.

 

[chickeneps]

Thanks for your reply.

No, that is NOT what linear means. A "linear equation" is of the form y= ax+ b for numbers a and b.

What I'm saying that for inputs of x, y is the same. So given a graph of x 0 to 1 and y 0 to 1, you get a an ascending 45 degree line, starting from the lower left corner to the upper right corner.

That's a linear slope. Forgive me if I am using the wrong technical language. Linear to me means output=input.

I have feeling you mean something more than this but I don't know what. Could you rephrase this? What do you mean by "harder equation" and "easier equation"? And I just realized you titled this "exponential function". So you mean something like $\displaystyle y= ab^x$? But the "slope" of such an equation varies from point to point.

My programming problem is that I will have inputs of 0 to 1, and I need a function where I'm either going to get the same number I gave it ("linear"), or values less than the input ("harder") or values more than the input ("easier"), while always starting with 0 and always ending with 1.

(The specific physical problem relates to how hard you hit a piano key. "Linear" means that if you hit it with 50% force, you get 50% impact. The goal is to make this variable, where you can hit the key softer yet get more impact on the string, or conversely you have to hit the key real hard to get the proper amount of impact. Regardless, if your velocity is 0, you get no string strike, and if your velocity is maximum ("1"), the string is struck with the maximum impact.)