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A. f(x) = x^2 -3x -10
Step-by-step explanation:
You can easily determine the correct equation by looking at a couple of table values.
The y-value for x=0 is -10, which is the constant in the quadratic formula. (eliminates choices B and C)
The y-value for x=1 is -12, which is the sum of coefficients in the quadratic formula. For choice A, the sum of coefficients is 1 -3 -10 = -12. For choice D, the sum of coefficients is 1 +3 -10 = -6. The value for choice D does not match the table, but the value for choice A does.
The function that represents the table is ...
f(x) = x^2 -3x -10
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Additional comments
If you don't have equations to test, and you want to write your own equation, there are a couple of ways to do that. One is to look at the differences of y-values, and the differences of those.
The sequence of y-values shown is ...
30, 18, 8, 0, -6, -10, -12
The sequence of differences of these numbers, starting with 18-30 = -12 is ...
-12, -10, -8, -6, -4, -2
The sequence of differences of these numbers is ...
2, 2, 2, 2, 2
The fact that they are constant tells you a quadratic (degree = 2) will describe the table.
We can find the coefficients a, b, c in the quadratic formula f(x) = ax^2 +bx +c as follows. First, we define d1 as the first of the sequence of differences (-12), and d2 as the value of the second differences (2). And we'll let x0 and y0 be defined as the first of the x-values (-5) and y-values (30).
The formulas for the coefficients (a, b, c) can be found from ...
a = (d2)/2 = 2/2 = 1
b = (d1) -a(2·(x0) +1) = -12 -1(2·(-5) +1) = -12 +9 = -3
c = y0 -x0(a·x0 +b) = 30 -(-5)(1·(-5) +(-3) = 30 -40 = -10
As we noted above, the y-value corresponding to x=0 is also this c value. When the table includes x=0 and x=1, those can be used to simplify the finding of the formula coefficients.
These computations tell us f(x) = x^2 -3x -10, as above.
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If you have to do this again, it might be useful to make a note of the expressions for 'a', 'b', and 'c' that we showed here, along with the definitions of the variables we used in them.
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Another way to find the formula for f(x) is to make use of a calculator or spreadsheet capable of quadratic regression. Give it three (x, y) pairs from the table, and it can tell you the formula coefficients. This is illustrated by the attachment.
