Friendship in Stardew Valley is measured by points. The more points you have with villagers, the better your relationship with them is. Villagers with better relationships will send you gifts in the mail, talk friendlier to you, and even give you recipes.

Above is a screenshot of the friendship menu from the game. Instead of seeing points, the player sees hearts. Each heart represents 250 points. Any remaining points will not partially fill a heart, so it is hard to tell in-game how many points a relationship with a villager has.

It is also worth noting that number of maximum hearts vary depending on the villager.

Regardless of the number maximum hearts, the method of increasing friendships is the same for all villagers.

Everyday, friendship with villagers decays. This decay is determined by the level of friendship. Also depending on this level, friendship will stop decaying when all hearts are full.

You can give gifts to villagers to increase or decrease friendship, how much so is determined by the gift type and the quality of the gift. The quality of the gift only affects liked and loved gifts, while neutral, disliked, and hated gifts are just their gift worth. Gifts can be given to villagers once per day and up to two gifts per week. Additionally, if maximum gifts were given in one week, then the friendship will receive a bonus of +10 points at the end of the week. Each villager has their own set of loved, liked, neutral, disliked, and hated gifts, which the wiki lists out for each.

There are two events that affect friendship levels: the Feast of the Winter Star and birthdays. A gift given on the Feast of the Winter Star has a multiplier of x5. A gift given on a villager's birthday has a multiplier of x8. These multipliers will enlarge the point value of the given gift, no matter if it is an increase or decrease in overall friendship.

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I decided to pick a situation to model for this project after viewing several different graphs using the algorithm I created. I am modelling talking to a non-romanceable villager once a day for the first 5 days in each week. The goal is to get the villager up to 10 hearts (or 2500 points), so the friendship doesn't decay anymore. According to my calculations, this would take 204 in-game days, or 1 year and 3.286 seasons.

After lots of adjusting, I reached this equation because:

p(x) = 18 * sin((x - 2.5) * (2 / 7)) + 12.285x + 15

p(x) calculates the total points in the relationship for the xth day since first talking to the villager.

r(x) = (36 / 7) * cos((x - 2.5) * (2 / 7)) + 12.285

r(x) calculates the rate of change of the total points in the relationship for the xth day since first talking to the villager.

a(x) = (-72² / 49) * sin((x - 2.5) * (2 / 7))

a(x) calculates the acceleration of the total points in the relationship for the xth day since first talking to the villager.

P(x) = (-63 / ) * cos((x - 2.5) * (2 / 7)) + 6.1425x² + 15x + c

The integral, P(x) doesn't really mean anything in the context of this problem, as the total number of points given by p(x) is really the only thing needed to compute the friendship in-game.

On the interval [0, 7] the derivative of the continuous function p(x), r(x), equals 0 at the x-values of 5.212 and 6.788. These correspond to the critical points (5.212, 90.726) and (6.788, 86.694).

On the interval [0, 7] the relative maximum is (5.212, 90.726). This maximum corresponds to the 5th day in the week, which represents the last day in the given week that the player talks to the villager. The relative minimums are (6.788, 86.694) and (0, 0.927). These minimums both represent the 7th day of the week, which is the last day in the given week that the friendship is decaying from the player not talking to the villager.

On the interval [0, 7] the second derivative of the continuous function p(x), a(x), equals 0 at the x-values of 2.5 and 6. This means that the inflection points are at (2.5, 45.712) and (6, 88.71).

Looking at the interval [0, 7], the inflection point x-values can be used to find the concavity of the function p(x).

I choose to change the linear growth parameter, as in the 12.285x. I changed this to 20x.

This made the graph increase faster than the original p(x). This change represents giving a greater point value to talking. It is also worth noting that the friendship doesn't decay with this new parameter, which adds to the graph increasing faster than the original. This can also be seen in the first derivative of the new function, where now there is no negative rate of change to be found. This means that there are no minimum and maximum extrema, as the first derivative doesn't cross the x-axis. The second derivative is the same with or without the parameter change, as the parameter in question is absent from both. This also means that the inflection points are spread out across the same x-values as before. The y-intercept is the same as before. The x-intercept is (-0.0988, 0), which is closer to (0, 0), but still out of the domain.