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Thread: cubic function graph

  1. #1
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    cubic function graph

    hi
    i got this :
    sketch, on a separate diagram, the curve with equation y = (x 2)^3 6(x 2)^2 + 9(x 2)

    and i dont know how, but i should get this
    cubic function graph-untitled.png
    but i end up with x^3 -12x^2 - 3x - 50
    any help appreciated
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  2. #2
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    Re: cubic function graph

    y = (x 2)^3 6(x 2)^2 + 9(x 2)
    factored form provides the best information for graphing by hand ...

    $y = (x-2)[(x-2)^2 - 6(x-2) + 9]$

    $y = (x-2)[(x-2)-3]^2$

    $y = (x-2)(x-5)^2$

    single zero at x = 2, zero of multiplicity two at x = 5 ... a "bounce" in the graph at (5,0).

    End behavior of a cubic function with a positive leading coefficient is $y \to \pm \infty$ as $x \to \pm \infty$
    Attached Thumbnails Attached Thumbnails cubic function graph-curve-sketch.jpg  
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  3. #3
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    Re: cubic function graph

    y = (x 2)^3 6(x 2)^2 + 9(x 2)
    you may also see this as a composite function ...

    $f(x) = x^3-6x^2+9x = x(x-3)^2$ ... single root at x=0 and double root at x=3

    $g(x) = x-2$

    $f[g(x)] = f(x-2) = (x-2)[(x-2)-3]^2 = (x-2)(x-5)^2$ the graph of $f$ shifted horizontally 2 units to the right.

    link has a good explanation for sketching ...

    Algebra - Graphing Polynomials
    Last edited by skeeter; Jul 9th 2017 at 08:53 AM.
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  4. #4
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    Re: cubic function graph

    i must be missing some basic knowledge because i get everything except how you got from [(x−2)^2−6(x−2)+9] to [(x−2)−3]^2
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  5. #5
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    Re: cubic function graph

    how you got from [(x−2)^2−6(x−2)+9] to [(x−2)−3]^2
    $(x−2)^2−6(x−2)+9$

    let $t = (x-2)$ ...

    $t^2 - 6t + 9 = (t-3)(t-3) = (t-3)^2 = [(x-2)-3]^2$
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