Answer:
c = 5
Step-by-step explanation:
You want to know the value of y when x=0, given the table of values shown.
2nd differences
We can work backward from the first three (x, y) pairs shown to find the pair corresponding to x=0.
x = 5, 4, 3, 2
y = 80, 53, 32, 17
First differences are ...
80 -53 = -27
32 -53 = -21
17 -32 = -15
And second differences are ...
-21 -(-27) = 6
-15 -(-21) = 6
These are constant, as expected for a quadratic function.
First differences
Using this same value (6) for the second differences, we can find the next first differences to be ...
-15 +6 = -9
-9 +6 = -3
Then the next values in the sequence will be ...
17 +(-9) = 8 . . . . . . y-value for x=1
8 +(-3) = 5 . . . . . . . y-value for x=0
The value of c is 5.
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Additional comment
We can also find the value other ways. One is to use the quadratic regression function of a calculator or spreadsheet. Another is to write equations for the coefficients a, b, c in the quadratic expression. The attachment shows the formula for the y-values is ...
y = 3x² +5 . . . . . . . c=5
The matrix row-reduction operation in the second attachment solves the three linear equations in the coefficient values associated with the first three table entries. It gives the same answer as above:
y = 3x² +5 . . . . . . . c=5
To find the coefficients of the quadratic function, we only need three pairs of values from the table. Using additional pairs just confirms that the relations we found are consistent with the table values.
