Answer:
both are Quadratic
Step-by-step explanation:
To determine if a table of function values represents a linear function or a quadratic function, you can often look at first- and second-differences of the dependent variable.
The x-values are evenly spaced in each case, so the question can be answered by looking at differences in y-values. The set of first differences is the result of subtracting a given column from the next. The set of second differences is found by doing the same, using the first differences.
Table 1
First differences; 21 -11 = 10, 35 -21 = 14, 53 -35 = 18, 75 -53 = 22
Second differences: 14 -10 = 4, 18 -14 = 4, 22 -18 = 4
The second differences all have the same value: 4. That means the function is 2nd degree, quadratic.
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Table 2
First differences: 10 -12 = -2, 9 -10 = -1, 9 -9 = 0, 10 -9 = 1
Second differences: -1 -(-2) = 1, 0 -(-1) = 1, 1 -0 = 1
The second differences all have the same value: 1. That means the function is 2nd degree, quadratic.
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Additional comment
The level at which differences are the same tells you the degree of the polynomial that will match the given function points. If first differences are constant, then the function is 1st degree: linear. Here, second differences are constant, so the functions are 2nd degree: quadratic. If you have to go to 4th differences to get constant values, then the table will require a 4th-degree polynomial to match its values. This presumes the x-values are all evenly-spaced.
If the x-values are not evenly spaced, then determining the degree of the function can be done by graphing and/or using regression formulas. Various interpolation polynomials are available for matching the points exactly.
