What is the equation, in standard form, of a parabola that contains the following points?
(2,0), (3, 2), (4,6)

  f(x) = x² -3x +2

Step-by-step explanation:

We know from the first point that (x -2) is a factor of the polynomial, so we can write the other factor as (ax+b). Filling in the values from the other two given points, we have ...

  f(x) = (x -2)(ax +b)

  f(3) = (3 -2)(3a +b) = 2

  f(4) = (4 -2)(4a +b) = 6

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From the first of these, ...

  3a + b = 2

Dividing the second by 2, we have ...

  4a + b = 3

Subtracting the first of these equations from the second gives ...

  (4a +b) -(3a +b) = (3) -(2)

  a = 1

Using this in the first of the above equations, we have ...

  3·1 + b = 2

  b = -1

Then the factored form of the equation is ...

  f(x) = (x -2)(x -1)

Expanding this to standard form, we have ...

  f(x) = x² -3x +2