How do I sketch the graph of a polynomial and find end behaviors and x intercepts of following function? $x^4-3x^3-3x^2+11x-6$
How do I sketch the graph of a polynomial and find end behaviors and x intercepts of following function? $x^4-3x^3-3x^2+11x-6$ ? I can tell I a factorization of this polynomial to graph it correctly, but I'm just really stuck on how to solve the problem. I have the basic idea of the terms asymptote and the basic step, but I'm just confused with this problem and I want to check if I'm doing it right as I solve it.
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$\begingroup$Note that $$x^4-3x^3-3x^2+11x-6 = (x-1)^2(x-3)(x+2)$$
You can easily find the x-intercepts and sketch the graph.
Note that polynomials are continuous and smooth functions without any asymptotes.
The end behavior is found by the highest degree and its coefficient.
Make sure to evaluate your function at points between the x-intercepts to find out the sign of your function in theses intervals.
$\endgroup$ 2 $\begingroup$Normally the way is to do a complete discussion of the function:
$f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^4-3x^3-3x^2+11x-6$.
This includes looking for symmetries, calculating the roots, the extrem points and turning points. You need the first three derivatives to do this.
For the roots you have to calculate
$f(x)=0\Leftrightarrow x^4-3x^3-3x^2+11x-6=0$
You have a polynomial of degree 4. The normal way is to do a long division. For that you need to guess a root. Possible candidates are the divisors of $-6$ which are $\{\pm 1, \pm 2, \pm 3, \pm 6\}$. With that you reduce the polynomial until you can use a formula (Like the pq-formula) to solve the remaining equation.
After that you might calculate the extrem points. For that you have to calculate $f'(x)=0$. Then you check if it is a maximum or minimum by checking if $f'(x)<0$ (maximum) or $f'(x)>0$ (maximum).
If $f'(x)=0$ you might get a saddle point.
For turning points you have to calculate $f''(x)=0$ and you have to check if $f'''(x)\neq 0$.
After that you have possible 4 roots, 3 extrema and 2 turning points. Dont forget to calculte the f(x) value. Draw them into a graph and scetch the graph. For the drawing it helps to mark the extrema if you get an maximum or minimum by drawing a "sad mouth (" for a maximum and a "happy mouth" for a minimum. [Of course turned by 90°... This is also just an explaination I just made up...]
For the asymptotic behaviour you might just "see" it after you scetched the graph or just calculate f(1000) or f(-1000) with an calculator to get an idea and then try to verify it with additional thoughts.
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