1. ## 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

but i end up with x^3 -12x^2 - 3x - 50
any help appreciated

2. ## 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 $x \to \pm \infty$

3. ## 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

4. ## 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

5. ## 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 $t = (x-2)$ ...

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