Given that f(-1)=3, f(2)=4, f(1)=1 find the quadratic function f(x)

We'll write the expression of the quadratic
function:

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

f(-1)=3

We'll substitute x
by -1 in the expression of the quadratic:

f(-1)=a*(-1)^2 +
b*(-1) + c=a-b+c

a-b+c=3
(1)

f(2)=4

f(2)=a*(2)^2
+ b*(2) + c

4a + 2b + c = 4
(2)

f(1)=1

f(1)=a*(1)^2
+ b*(1) + c=a+b+c

a+b+c=1
(3)

We'll add (1) +
(3):

a-b+c+a+b+c=3+1

We'll
eliminate and combine like terms:

2a + 2c =
4

We'll divide by 2:

a + c = 2
(4)

We'll multiply (1) by
2:

2a - 2b + 2c = 6 (5)

We'll
add (5) + (2):

2a - 2b + 2c + 4a + 2b + c = 6 +
4

We'll eliminate and combine like
terms:

6a + 3c = 10 (6)

We'll
multiply (4) by -3:

-3a - 3c = -6
(7)

We'll add (6) + (7):

6a +
3c -3a - 3c = 10 - 6

3a =
4

We'll divide by
3:

a =
4/3

We'll substitute a in 6a + 3c =
10.

6*(4/3) + 3c = 10

8 + 3c =
10

3c = 10-8

3c =
2

c =
2/3

We'll substitute a and c in
(1):

4/3 - b + 2/3 = 3

2 - b =
3

We'll subtract 2 both
sides:

-b = 3-2

-b =
1

b =
-1

The
quadratic funation
is:

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