SOLUTION: A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2.If f(1)=1, and f(-2) =-2, find f(-1) and write the complete function.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2.If f(1)=1, and f(-2) =-2, find f(-1) and write the complete function.       Log On


   

Question 930999: A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2.If f(1)=1, and f(-2) =-2, find f(-1) and write the complete function.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
A cubic polynomial function:
+f%28x%29+=ax%5E3%2Bbx%5E2%2Bcx%2Bd
given:
has leading coefficient +-2+=>highlight%28a=-2%29
so far we know +f%28x%29+=-2x%5E3%2Bbx%5E2%2Bcx%2Bd
and intercepts the y-axis at 2: => highlight%28d=2%29=> +f%28x%29+=-2x%5E3%2Bbx%5E2%2Bcx%2B2
If f%281%29=1 => +f%281%29+=-2%281%29%5E3%2Bb%281%29%5E2%2Bc%281%29%2B2=1
=>+-2%2Bb%2Bc%2B2=1
=>+b%2Bc=1.......eq.1

and f%28-2%29+=-2 => +f%28-2%29+=-2%28-2%29%5E3%2Bb%28-2%29%5E2%2Bc%28-2%29%2B2=-2
=> +-2%28-8%29%2Bb%284%29%2Bc%28-2%29%2B2=-2
=>16%2B4b-2c%2B2=-2
=>+4b-2c=-20.......eq.2
solve the system:
+b%2Bc=1.......eq.1
+4b-2c=-20.......eq.2
_________________________
+b%2Bc=1.......eq.1 ...solve for b
+b=1-c...substitute in eq.2
+4%281-c%29-2c=-20.......eq.2...solve for c
+4-4c-2c=-20
+4-6c=-20
+4%2B20=6c
+24=6c
+24%2F6=c
highlight%28c=4%29
find b
+b=1-c
+b=1-4
highlight%28+b=-3%29
the complete function is:
+f%28x%29+=-2x%5E3-3x%5E2%2B4x%2B2

find f%28-1%29 :
+f%28-1%29+=-2%28-1%29%5E3-3%28-1%29%5E2%2B4%28-1%29%2B2
+f%28-1%29+=-2%28-1%29-3%281%29%2B4%28-1%29%2B2
+f%28-1%29+=2-3-4%2B2
+f%28-1%29+=-3

check:
+f%28-2%29+=-2%28-2%29%5E3-3%28-2%29%5E2%2B4%28-2%29%2B2
+f%28-2%29+=-2%28-8%29-3%284%29%2B4%28-2%29%2B2
+f%28-2%29+=16-12-8%2B2
+f%28-2%29+=4-8%2B2
+f%28-2%29+=6-8
+f%28-2%29+=-2...as given