How do you find the important points to graph $f (x) = x^{3} - 9 x^{2} + 27 x - 26$ $f(x) = x^3 - 9x^2 + 27x - 26$ ?

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How do you find the important points to graph $f (x) = x^{3} - 9 x^{2} + 27 x - 26$ $f(x) = x^3 - 9x^2 + 27x - 26$ ?

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Answer 1 · 2017-05-12T22:13:52

See explanation...

Explanation:

Given:

$f (x) = x^{3} - 9 x^{2} + 27 x - 26$ $f(x) = x^3-9x^2+27x-26$

I can see that this is quite similar to ${(x - 3)}^{3}$ $(x-3)^3$ . Let's expand that:

${(x - 3)}^{3} = x^{3} + 3 (x^{2}) (- 3) + 3 (x) {(- 3)}^{2} + {(- 3)}^{3}$ $(x-3)^3 = x^3+3(x^2)(-3)+3(x)(-3)^2+(-3)^3$

${(x - 3)}^{3} = x^{3} - 9 x^{2} + 27 x - 27$ $color(white)((x-3)^3) = x^3-9x^2+27x-27$

So we find:

$f (x) = {(x - 3)}^{3} + 1$ $f(x) = (x-3)^3+1$

Now the sum of cubes identity can be written:

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ $a^3+b^3 = (a+b)(a^2-ab+b^2)$

So we find:

$f (x) = {(x - 3)}^{3} + 1$ $f(x) = (x-3)^3+1$

$f (x) = {(x - 3)}^{3} + 1^{3}$ $color(white)(f(x)) = (x-3)^3+1^3$

$f (x) = ((x - 3) + 1) ({(x - 3)}^{2} - (x - 3) + 1)$ $color(white)(f(x)) = ((x-3)+1)((x-3)^2-(x-3)+1)$

$f (x) = (x - 2) (x^{2} - 6 x + 9 - x + 3 + 1)$ $color(white)(f(x)) = (x-2)(x^2-6x+9-x+3+1)$

$f (x) = (x - 2) (x^{2} - 7 x + 13)$ $color(white)(f(x)) = (x-2)(x^2-7x+13)$

Note that $f (0) = - 26$ $f(0) = -26$

So we have:

$f (x)$ $f(x)$ has $x$ $x$ intercept $(2, 0)$ $(2, 0)$
$f (x)$ $f(x)$ has $y$ $y$ intercept $(0, - 26)$ $(0, -26)$
$f (x)$ $f(x)$ is like $y = x^{3}$ $y = x^3$ but shifted right by $3$ $3$ units and up by $1$ $1$ unit.

Putting this all together we find $f (x)$ $f(x)$ looks like this:

graph{(y-(x^3-9x^2+27x-26))(6x^2+(y+26)^2-0.06)(6(x-2)^2+y^2-0.06)(6(x-3)^2+(y-1)^2-0.06) = 0 [-7.375, 12.625, -34, 18]}

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How do you find the important points to graph $f (x) = x^{3} - 9 x^{2} + 27 x - 26$ $f(x) = x^3 - 9x^2 + 27x - 26$ ?

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Explanation:

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How do you find the important points to graph f(x)=x3−9x2+27x−26f(x) = x^3 - 9x^2 + 27x - 26?

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How do you find the important points to graph $f (x) = x^{3} - 9 x^{2} + 27 x - 26$ $f(x) = x^3 - 9x^2 + 27x - 26$ ?