Answer: y = x^3+2x^2+3
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Step-by-step explanation:
If you plug in x = 0 and y = 3, then the original equation boils down to c = 3. The x = 0 cancels out pretty much everything except that c term.
We have y=ax^3+bx^2+c update to y=ax^3+bx^2+3
Now plug in x = 1 and y = 6 from the point (1,6)
y=ax^3+bx^2+3
6=a(1)^3+b(1)^2+3
6 = a+b + 3
a+b = 6-3
a+b = 3
Now plug in the coordinates of (-1,4) which are x = -1 and y = 4
y=ax^3+bx^2+3
4=a(-1)^3+b(-1)^2+3
4 = -a + b + 3
-a+b = 4-3
-a+b = 1
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Here are the two equations we're interested in
a+b = 3
-a+b = 1
We'll add these equations straight down. The 'a' terms cancel. The b terms add to 2b. The right hand sides add to 3+1 = 4
We're left with 2b = 4 and it solves to b = 2
Use any of the equations involving 'a' and b to determine that a = 1
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To summarize we have
a = 1
b = 2
c = 3
This means y=ax^3+bx^2+c updates fully to y=1x^3+2x^2+3 which simplifies to y = x^3+2x^2+3 and it's the final answer.
Visual verification is shown below. Desmos is a graphing tool I use all the time, as well as GeoGebra. Both are free apps.