How to implement a gradual increase in the number?

Question

PHP

I don't know how to better phrase the question in the title.
I'm not very good at math, so I have to ask for help.
I'll try to describe it in more detail here. There is such a thing as a geometric progression and everyone knows about it, but it's too simple for my task.
There are two numbers: X and Y
X = 1000
Y = 0.1
I want that every time X increases by 100, Y increases, but its increase will be the greater, the more X. For example:
X = 1000, Y = 0.1
X = 1100, Y = 0.12, +0.02
X = 1200, Y = 0.144, +0.024
X = 1300, Y = 0.175, +0.031
And so on. Each time the increase is stronger.
Honestly, I want to pick up the values ​​​​by eye so that with X = 5000, Y would be equal to ~ 12-15.
It would be nice to have some coefficient in the formula to control the rate of increase in the number.
How to do it right?

Answer 1

lerp

Answer 2

$x = 1; $y = $x/20;
$x = $x+$y;

Roughly speaking, so x will always grow by 5% of itself, the more x - the more it grows or is it nonlinearly necessary?

Answer 3

You need a quadratic (or other power) dependence: y = x^2.
The derivative of this expression will be the growth rate of the function:y′ = 2x

Answer 4

Method of finite differences.
0.100; 0.120; 0.144; 0.175 etc. - from below, attribute the difference between the members.
0.020; 0.022; 0.031; etc. - and again from below, attribute the difference.
0.002; 0.009; etc. - and so on
until the same numbers appear in the line.
In this case, the number of rows is the power of the formula a(n) * xⁿ + ... a₃ * x³ + a₂ * x² + a₁ * x + a₀
This formula, as I understand it, you are looking for. But this method is only for p̲o̲l̲i̲n̲o̲m̲i̲a̲l̲b̲n̲o̲g̲o̲ expression, not for exponential, trigonometric or any other.
There, the method describes how to extract the coefficients a₁, a₂... from the calculated triangle, but I don’t remember how.

Answer 5

deltaY = exp((x-900)/1480) - exp((x-1000)/1480)

1000	+0,069902548	0,079902548
1100	+0,074788914	0,154691462
1200	+0,08001685	0,234708312
1300	+0,085610231	0,320318543
...
4800	+0,911115494	12,95519682
4900	+0,974804788	13,93000161
5000	+1,042946127	14,97294774

Answer 6

An example of a task implementation in PHP:

$x = $result = [];

$minY = 0.1;
$minX = 1000;
$maxX = 5000;
$dX = 100;

for( $n = $minX; $n <= $maxX; $n += $dX)
    $x[] = $n;

foreach ($x as $val)
    $result[$val] = ($minY * ($val ** 3) )/($minX ** 3);

echo '<pre>';
print_r($x);
print_r($result);

I hope you figure out what's what :)