Let the function be f(x) = ax^2 + bx + c

Given the points ( -1, 4) , (1, 2), and ( 2,7) are on the curve f(x).

Let us substitute :

==> f(-1) = a(-1)^2 + b(-1) + C = 4

==> a - b + c = 4.................(1).

==> f(1) = a(1^2) +b(1) + C = 2

==> a + b + c = 2...............(2).

==> f(2) = a(2^2) + b(2) + C = 7

==> 4a + 2b + C = 7.............(3).

Now we have a system of 3 equations and 3 variables.

We will use the elimination method to solve.

Let us add (1) and (2).

==> 2a+ 2c = 6

==> a + C = 3 ...............(4).

Now we will add 2*(1) and (3)

==> 2a-2b + 2c = 8

==> 4a+2b + C = 7

==> 6a + 3c = 15

==> 2a + c = 5............(5)

Now subtract (4) from (5).

==> a = 2

==> c = 1

==> b = -1

==>** f(x) = 2x^2-x + 1**

Let the quadratic equation be ax^2 + bx + c.

Now f(-1)=4

f(1)=2

f(2)=7

Substitute x = 1 in ax^2 + bx + c

=> a + b + c = 2...(1)

Substitute x = -1 in ax^2 + bx + c

=> a - b + c = 4...(2)

Substitute x = 2 in ax^2 + bx + c

=> 4a + 2b + c = 7...(3)

(1) - (2)

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

=> 2b = -2

=> b = -1

Substitute b = -1 in (2)

=> a + 1 + c = 4 => a + c = 3

Substitute b = -1 in (3)

=> 4a -2 + c = 7 => 4a + c = 9

Now, a + c = 3 => a = 3 - c

Using this in (3)

4(3 - c) + c = 9

=> 12 - 4c + c = 9

=> -3c = 9-12

=> c = -3 / -3 = 1

As a = 3 - c = 3 - 1 = 2

Therefore a = 2, b = -1 and c = 1

**So the quadratic is 2x^2 - x + 1**

To determine the quadratic f(-1)=4, f(1)=2, f(2)=7 .

Let f(x) = ax+by +c.

Then f(-1) = a(-1)^2+b(-1)+c = -4. Or a-b+c = 4...(1).

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

f(2) = a*2^2+b*2+c = 7. Or 4a+2b+c = 7....(3).

Eq(1)+eq (2): 2(a+c) = 6.

So a+c = 3..........(4).

Eq(1)*2+ eq(4): 6a+ 3c = 4*2+7 = 15.

So 3(2a+c) = 115.

2a+c = 15/3 = 5.

2a+c = 5.....(5).

Eq(5) - eq(4): a = 5-3 = 2.

Put a=2 in eq(5): 2*2+c= 5. So c= 5-4 = 1.

Put a = 2,c= 1 in eq(2): 2+b+1 = 2. So b = 2- 2-1 = -1.

Therefore a= 2, b= -1 and c = 1.

So f(x) = ax^2+bx+c = **2x^2-x+1**.

We'll write the equation of the quadratic function:

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

This function is determined if and only if the coefficients a,b,c, are determined.

We'll impose the constraints given by enunciation:

f(-1) = 4

We'll substitute x by -1:

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

a - b + c = 4 (1)

f(1) = 2

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

a + b + c = 2 (2)

f(2) = 7

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

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

4a + 2b + c = 7 (3)

We'll add (1) + (2):

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

We'll combine and eliminate like terms:

2a + 2c = 6

We'll divide by 2:

a + c = 3 (4)

We'll add 2*(1) + (3):

2a - 2b + 2c + 4a + 2b + c = 8 + 7

We'll combine and eliminate like terms:

6a + 3c = 15

We'll divide by 3:

2a + c = 5 (5)

We'll subtract (4) from (5):

2a + c - a - c = 5 - 3

**a = 2**

2 + c = 3

c = 3 - 2

**c = 1**

2 - b + 1 = 4

3 - b = 4

**b = -1**

**The quadratic function is:**

**f(x) = 2x^2 - x + 1 **