Question 1119738: Mary's Towel Company recorded the following data on the number of towels sold (x) and the company's total profit (y).
Number of towels sold 2 , 3 , 4 , 5 , 6 , 7 , 8
Total profit. $7 , $16 , $27, $40 , $55 , $72 , $91
If the company creates a scatter plot of the data, what will be the equation of the curve of best fit?
A.f(x)= x^2+4x-5
B.f(x)= x^2+5x-4
C.f(x)= 2x^2-4x-5
D.f(x)= 2x^2-5x-4
Found 2 solutions by rapture1965, greenestamps:
Answer by rapture1965(18)   (Show Source):
You can put this solution on YOUR website! Let us look at the first two values 2 and 7.
Let x = 2.
If I plug 2 in for x in the first equation, will I get 7 for f(x) noting that
f(x) is the same as y?
Well, let us see what happens.
f(x) = x^2 + 4x - 5
Replace every x you see with 2.
f(2) = (2)^2 + 4(2) - 5
f(2) = 4 + 8 - 5
f(2) = 12 - 5
f(2) = 7
So far so good. We just learned that when x is 2, y is 7.
Let us try the next x-value, which is 3.
If I let x = 3 and get 16, we can stop right there. There will be no need
to try the other equations on the list.
Replace every x you see with 3.
f(3) = (3)^2 + 4(3) - 5
f(3) = 9 + 12 - 5
f(3) = 21 - 5
f(3) = 16
What can we conclude?
The only equation from the ones listed that yields 7 and 16 after plugging 2 and
3 respectively is choice A.
Answer: Choice A
Answer by greenestamps(13216)  (Show Source):
You can put this solution on YOUR website!
Note first that "scatter plot" and "curve of best fit" suggest that the data is not completely well behaved. But in fact these numbers ARE very well behaved. The values of x are consecutive integers, and the increase in the y values changes by 2 from one pair to the next.
Second, note that you can find the right answer without using any formal mathematical methods, by simply trying the given x values in each of the answer choices and finding the choice that gives the right y value for every x value.
But a little knowledge about the method of finite differences will make it easy to find the correct answer using formal mathematics.
With the method of finite differences, you make a display of the y values, and the differences between successive y values, and the differences between those differences, and so on, until you find a row of constant differences. For these numbers, it looks like this:
7 16 27 40 55 72 91
9 11 13 15 17 19
2 2 2 2 2
The constant 2 in the second row of differences means the data are defined by a polynomial of degree 2; it also means the leading coefficient of the polynomial is 2/(2!) = 2/2 = 1. So formal mathematical methods can tell you the answer is either A or B.
You could go on from there with other formal mathematical methods to determine which of A or B is the right polynomial; but of course again it is easier just to plug in the given x values and see which of A or B gives the right y values.
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