`f(x)=3x^2+6x+7`

Given that f(x) can be written in the form `A(x+B)^2+C`

where A, B and C are rational numbers.

  • Find the value of A, B and C
  • Hence/Otherwise, find:
  •                                         a) the value of x for which`f(x)`

                                                is a minimum

                                            b) The minimum value of `f(x)`

    Step by step answers

    Expert Answers

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    1)  The number beside `x^2` is A, so:

    `f(x)=3(x+B)^2+C`

    2)  Factor out A.  That is, divide everything by 3:

    `f(x)=3(x^2+2x+7/3)`

    3)  The number that is next to `x` is double B

    So, to find B, take 2 and cut in half

    So B =1

    `f(x)=3(x+1)^2+C`

    4)  There are a few...

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    1)  The number beside `x^2` is A, so:

    `f(x)=3(x+B)^2+C`

    2)  Factor out A.  That is, divide everything by 3:

    `f(x)=3(x^2+2x+7/3)`

    3)  The number that is next to `x` is double B

    So, to find B, take 2 and cut in half

    So B =1

    `f(x)=3(x+1)^2+C`

    4)  There are a few ways to find C.  Here is one of them:

    You want

    ``

    ``

    ``

    ``

    ``

    `4=C`

    So:

    `f(x)=3(x+1)^2+4`

    One of the reasons this is useful, is it immediately tells you the minimum of the parabola.  (If A is negative, it is an upside-down parabola, and instead the parabola has a maximum)

    The minimum value occurs at `x=-B`

    So, for this problem, the minimum occurs at `x=-1`

    Plug in to find what the minimum value is:

    `f(-1)=3(-1+1)^2+4=0+4=4`

    That is, the minimum value is just C

    To summarize:

    If `f(x)=A(x+B)^2+C`

    then the minimum occurs at `x=-B` , and the minimum is `C`

    (if A is negative, then the maximum occurs at -B, and the maximum is C)

    So, for us, the value of `x` for which `f(x)` is a minimum is -1

    and the minimum value of `f(x)` is 4

    Approved by eNotes Editorial Team