Given the following three points, find by hand the quadratic function they represent.

(0,6), (2,16), (3, 33)

- f(x)=−4x2−3x+6
- f(x)=4x2+3x+6
- f(x)=−4x2+21x+6
- f(x)=4x2−3x+6

I am struggling a lot with this one. But i think its A

1 answer

Quadratic function:

y = a x² + b x + c

In this case:

for x = 0 , y = 6

y = a x² + b x + c

6 = a ∙ 0² + b ∙ 0 + c

6 = 0 + 0 + c

6 = c

c = 6

Your quadratic function become:

y = a x² + b x + 6

for x = 2 , y = 16

y = a x² + b x + 6

16 = a ∙ 2² + b ∙ 2 + 6

16 = 4 a + 2 b + 6

Subtract 6 to both sides

10 = 4 a + 2 b

4 a + 2 b = 10

Divide both sides by 2

2 a + b = 5

for x = 3 , y = 33

y = a x² + b x + 6

33 = a ∙ 3² + b ∙ 3 + 6

33 = 9 a + 3 b + 6

Subtract 6 to both sides

27 = 9 a + 3 b

9 a + 3 b = 27

Divide both sides by 3

3 a + b = 9

Now you must solve system:

2 a + b = 5 , 3 a + b = 9

2 a + b = 5
-
3 a + b = 9
_________
- a = - 4

Multiply both sides by - 1

a = 4

Replace this value in equation:

2 ∙ 4 + b = 5

8 + b = 5

Subtract 8 to both sides

b = - 3

So:

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

y = a x² + b x + c

y = 4 x² - 3 x + 6