Question 1209647: Suppose f(x) is a quadratic function such that f(1) = -24, f(4) = 10, and f(3) = 60.
Determine the value of f(-1).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine f(-1):
1. **General form of a quadratic function:**
f(x) = ax² + bx + c
2. **Set up a system of equations using the given information:**
* f(1) = a(1)² + b(1) + c = a + b + c = -24
* f(4) = a(4)² + b(4) + c = 16a + 4b + c = 10
* f(3) = a(3)² + b(3) + c = 9a + 3b + c = 60
3. **Solve the system of equations:**
There are several ways to solve this system. One approach is to use elimination or substitution. Here's one way:
* Subtract the first equation from the second:
15a + 3b = 34
* Subtract the first equation from the third:
8a + 2b = 84, which simplifies to 4a + b = 42
* Solve for b in terms of a from the simplified third equation:
b = 42 - 4a
* Substitute this expression for b into the equation from subtracting the first two:
15a + 3(42 - 4a) = 34
15a + 126 - 12a = 34
3a = -92
a = -92/3
* Substitute a back into the equation for b:
b = 42 - 4(-92/3)
b = 42 + 368/3
b = (126 + 368)/3
b = 494/3
* Substitute a and b back into the first equation to solve for c:
(-92/3) + (494/3) + c = -24
402/3 + c = -24
134 + c = -24
c = -158
4. **Write the quadratic function:**
f(x) = (-92/3)x² + (494/3)x - 158
5. **Calculate f(-1):**
f(-1) = (-92/3)(-1)² + (494/3)(-1) - 158
f(-1) = (-92/3) - (494/3) - 158
f(-1) = -586/3 - 158
f(-1) = -586/3 - 474/3
f(-1) = -1060/3
f(-1) = -353.3333...
Therefore, f(-1) = -1060/3 or approximately -353.33.
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