# Determine the quadratic if f(-1)=4, f(1)=2, f(2)=7 .Determine the quadratic if f(-1)=4, f(1)=2, f(2)=7 .

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

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

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

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

(1) - (2)

=> - 2b = 2

=> b = -1

Substitute in (2)

=> a + c = 3

Substitute in (3)

=> 4a + c = 9

4a + c - 4a - 4c = 9 - 12

=> -3c = -3

=> c = 1

Substitute b = -1 and c = 1 in (1)

=> a +1 + 1 = 4

=> a = 2

**The quadratic function is f(x) = 2x^2 - x + 1 = 0**

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 **