Comparing models to fit data example

christine works at a movie store in her hometown using the store's total selection she documented the price of each movie title and how many years it has been since it was featured in movie theaters she plotted the points below so let's see what's going on below here so let's see it looks like she there's two curves that she tries to fit I'm assuming we're going to read about in that in a second but these blue points are the data points so for example this data point right over here shows a movie that the title costs six dollars and it has been released for almost two years a little under two years this data point right over here this is a movie that has been released for looks like almost four years it looks like maybe three and three quarters years and it they're selling that looks like for a dollar or even a little bit less than a dollar so those are her data points so once again she documented the price of each movie title as a function as a function of how long it's been how many years it's been since it was featured in movie theaters she is looking for a function that models her data since the trend of the data is decreasing in convex and you see it here when it's decreasing it's definitely decreasing in convex it's opening it's opening upwards if you imagine a curve it looks like it's opening upwards a little bit like that so decreasing in convex she found a decreasing convex exponential model and a decreasing convex quadratic model so which of the following better which of the following functions better fits the data so function a this is an exponential this is the one in green right over here and function B this one right over here is a quadratic and you can see this one in purple and so which of one of those better fits the data and so if we look at if we look at what's going on here the green function the exponential one most of the data points for any given duration form you know how long that titles been out it looks like it's consistently under estimating that it's always you know it's the models guess for the or the model what the model would say the price is is always it at least 4 except for only as essentially except for only one data point right over here for all of these other data points it's under estimating what the price would be the purple model or the purple function right over here it is it has a more of a balance between overestimating right over here and so it's overestimating by a little bit and under estimating and it's under estimates are closer and it's over estimates are closer than this a green model so I would say that function B is definitely definitely a better a better model use the function of best fit so we're going to say function B to predict the price of a movie that was featured in theaters 5.5 years ago round your answer to the nearest cent so five point five years ago that's going to be right over here we're going to go to function B which is this purple one so it's going to be you know it's going to be under a dollar but we want to get something to the nearest cent so let's actually use the the actual definition of the function so this is price as a function of how long the movie has been released where X is a how long has it been released and Y is its price so let's just let's just if X is five point five let's figure out what Y is going to be so it's going to be so Y is going to be equal to zero point five times x squared so X is five point five five point five squared all right so then we have minus 5 times X again so minus five times five point five and then we have plus 13 and what does that get us that gets us 60 two and a half cents if we round our answer to the nearest cent that's going to be 63 cents and we got it right