Answer:
f(x) = -x² -2x -4
Explanation:
You want a quadratic function that fits the points (-3, -7), (0, -4), and (2, -12).
Equations
A straightforward way to find the quadratic function is to write equations for the values of the coefficients.
f(x) = ax² +bx +c
f(-3) = -7 = a(-3)² +b(-3) +c = 9a -3b +c
f(0) = -4 = a(0)² +b(0) +c = c . . . . . . c = -4
f(2) = -12 = a(2)² +b(2) +c = 4a +2b +c
Solution
These can be solved by a variety of methods. The second attachment shows one of them. It tells us (a, b, c) = (-1, -2, -4).
Since we know c = -4, we can write this as 2 equations in 'a' and 'b'.
9a -3b = -3
4a +2b = -8
Removing common factors gives the standard-form equations ...
3a -b = -1
2a +b = -4
Adding these, we have ...
5a = -5
a = -1
So, ...
2(-1) +b = -4
b = -2
The quadratic function is ...
f(x) = -x² -2x -4
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Additional comment
The quadratic regression function of a calculator or spreadsheet can also find the equation for you. This is also shown in the second attachment.
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