The parabola f(x) = ax^2+bx+c passes through the points (1,6), (0,3), and (-1,4). Find the values of a, b, & c

Answer 1

Since the x values -1, 0, 1 are evenly spaced, you can use differences

4 3 6 # the values at -1, 0, 1 respectively
-1 3 # differences 3 - 4, 6 - 3
4 # difference 3 - (-1)

Take that 4, -1, 4 down the left hand side and you get

f(y) = (top / 0!) + (middle / 1!) y + (bottom / 2!) y (y - 1)
= 4 - y + 2 y (y - 1)
= 2 y^2 - 3 y + 4
which would be the solution if the initial x had been 0 with a step size of 1. Well, the step size is 1, but instead of starting at 0 it started at -1. So y = 0 corresponds to x = -1 and so x = y - 1, or y = x + 1, resulting in

f(x) = 2 (x + 1)^2 - 3 (x + 1) + 4
= 2 x^2 + x + 3

[If the step size had not been 1, the more general form would be y = ax + b, with a and b such that it maps the first and second x's to 0 and 1.

Check:
f(-1) = 2 - 1 + 3 = 4
f(0) = 3
f(1) = 2 + 1 + 3 = 6

With a cubic term you would have need the next (x, f(x)) pair in sequence and there would have been another row at the bottom, its numerical value would be the coeeficient of y (y - 1) (y - 2) / 3!, etc.

a = 2, b = 1, c = 3

Answer 2

X is the first coordinate. F(x) is the second one. Plug in the values for each set and you get three equations with three unknowns.

So, for the point(1,6) you get:

6=(1^2)a +(1)b +c = a + b + c

and so on. Solve the equations.

I am not doing your homework for you!

Answer 3

f(x) = ax^2 + bx + c

6 = a(1)^2 + b(1) + c
6 = a + b + c

3 = a(0)^2 + b(0) + c
3 = 0 + 0 + c
c = 3

4 = a(-1)^2 + b(-1) + c
4 = a - b + c

so now we have

a - b + c = 4
c = 3
a + b + c = 6

so now we have

a - b + 3 = 4
a + b + 3 = 6

a - b = 1
a + b = 3

the b's cancel out

2a = 4
a = 2

a - b = 1
2 - b = 1
-b = -1
b = 1

a = 2
b = 1
c = 3

Answer 4

JC is correct. The answer is A=2, B=1, C=3.
You do the math.

Answer 5

substitute values of x,y in the equation to get 3 simultaneous linear equation with three variable.

6=a+b+c
3=c
4=a-b+c

a+b=3
a-b=1
hence a=2 and b=1
so the solution for (a,b,c) is (2,1,3)

Answer 6

a = 2
b = 1
c = 3