Answer:
y = 1/2x^2 +6
Explanation:
The quadratic regression function of a calculator or spreadsheet is useful for this purpose. It shows you the equation is ...
y = 1/2x^2 +6
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Alternate solution
Another way to approach this is to look at first and second differences of the y-values.
Sequence of y-values:
6, 6.5, 8, 10.5, 14
Sequence of first differences:
0.5, 1.5, 2.5, 3.5
Sequence of second differences:
1, 1, 1, 1
The leading coefficient of the quadratic is half the value of the second differences: a = 1/2. The constant in the quadratic is the y-intercept, the value when x=0. The first point gives that as 6.
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Now, you can check some other point to see what the linear term coefficient might be:
y = 1/2x^2 +bx +6
For x=2, this is ...
8 = 1/2(2^2) +b(2) +6 = 8 +2b . . . . . substitute (x, y) = (2, 8), simplify
0 = 2b ⇒ b = 0
Then the quadratic is ...
y = 1/2x^2 +6
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Additional comment
You may not know the relation between second differences and the leading coefficient. The most general way to solve this is to write three equations in the unknown coefficients. Since y has integer values for x=0, 2, 4, you may want to use these. For (x, y) = (0, 6), (2, 8), and (4, 14), you have ...
y = ax² +bx +c
- 6 = 0a +0b +c
- 8 = a(2²) +b(2) +c
- 14 = a(4²) +b(4) +c
After finding c=6 in the first equation, you have two equations in 2 unknowns:
- 2 = 4a +2b ⇒ 2a +b = 1
- 8 = 16a +4b ⇒ 4a +b = 2
The solution to these is a=1/2, b=0, as above.