Question

How can I write down the function from the graph?

Answer 1

See explanation...

Explanation:

It looks like the graph passes through data points with approximate values:

`((0, 0), (5, 25), (10, 50), (15, 80), (20, 110), (25, 150), (30, 200), (35, 275))`

It is not clear from the graph whether these points are intended to be joined with straight line segments, indicating a piecewise linear function, or with a curve.

Since the sample points are at regular `x` intervals, we can use a method of differences to attempt to spot if they follow a polynomial pattern.

Write down the sequence of `y` values:

`0, 25, 50, 80, 110, 150, 200, 275`

Write down the sequence of differences between consecutive terms:

`25, 25, 30, 30, 40, 50, 75`

Write down the sequence of differences of those differences:

`0, 5, 0, 10, 10, 25`

Write down the sequence of differences of those differences:

`5, -5, 10, 0, 15`

That does not look like a pattern to me. Instead it looks like the function is intended to consist of linear segments.

The slope of each segment is given by dividing the rise by the run `5`, to find slopes:

`5, 5, 6, 6, 8, 10, 15`

We can treat the first couple of pairs of segments as single segments of length `10` and write a piecewise linear function:

`f(x) = { (5x " if " 0 <= x < 10), (6x - 10 " if " 10 <= x < 20), (8x - 50 " if " 20 <= x < 25), (10x - 100 " if " 25 <= x < 30), (15x - 250 " if " 30 <= x <= 35) :}`

At each point where we start the next segment, we choose the offset to make the value of the function match the sample point. Then the slope we calculated takes care of the rest until the next segment.

Bonus

If you are not keen on these piecewise definitions, there is an alternative using the properties of `x+abs(x)`, which in plain form looks like this:
graph{x+abs(x) [-10.71, 9.29, -2.84, 7.16]}

We can use scaled and offset copies of this function added together to get our incrementally increasing slopes and define:

`f(x) = 5x+1/2(x-10+abs(x-10))+(x-20+abs(x-20))+(x-25+abs(x-25))+5/2(x-30+abs(x-30))`

graph{5x+1/2(x-10+abs(x-10))+(x-20+abs(x-20))+(x-25+abs(x-25))+5/2(x-30+abs(x-30)) [-5, 40, -35, 280]}