Designing a game score function with non-linear graph

Question

I am trying to create a scoring scale for a game. I want the scale to be non-linear to reflect the realness of the game. The score is determined by time, the less time you take, the greater score you get.

• If you take more than $5$ minutes to complete the challenge, your score is $0.$

• If you take $0$ seconds to complete the challenge, your score is $40.$

My current equation is linear: $$y = {\rm MAX\ SCORE} - \frac{{\rm MAX\ SCORE}}{{\rm TIME\ TO\ MIN\ SCORE}}\times {\rm time}$$

However, I want the score to accelerate more quickly towards $40$ the closer you get to 0. What equation can I use to do this?

Answer 1

You can solve this:$$f(x) = {a \over {1+x}} + b$$ $$s.t: f(0) = \text{MAX_SCORE and } f(\text{TIME_TO_MIN_SCORE}) = 0$$

which results to (for your case with times in minutes you can also solve it for seconds, ...):

$$f(x) = {{48 \over (x+1)} -8}$$

Update: $$f(0) = 40, f(5) = 0 \Rightarrow$$ $$a+b=40, a + 6\cdot b = 0 \Rightarrow$$ $$a=48, b = -8$$

I assumed your time is distributed in minutes, you can have your other assumptions, e.g for seconds it will be like:$$f(0)=40, f(300) = 0, ...$$

Answer 2

An easy approach is to do $score =maxscore (\frac {maxtime-t}{max time})^n$ where you choose $n$ as high as you like to make the score fall quickly at the start. Alternately, $score=0.1maxscore(\frac {maxtime}{0.1maxtime+t})$, again adusting the 0.1 to your liking.