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Cubic function is $f(x) = ax^3 + bx^2 + cx + d , a\not = 0$ . I'm looking for a general way to plot it quickly . My mean is the plot it just by looking at the coefficients (i.e. $a , b ,c ,d$) . I don't want to be very accurate , just grasping the overview .

S.H.W
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    i think this will help you http://www.onlinemathlearning.com/cubic-function.html – Dr. Sonnhard Graubner Jul 25 '17 at 10:35
  • @Dr.SonnhardGraubner Thank you but it is useful in the special cases . – S.H.W Jul 25 '17 at 10:38
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    the inflection point at $\frac{-b}{3a}$, $d$ - ordinate of y-axis intersection, sign of $b^2-3ac$-gives you amount of horizontal tangents, knowing $a$ negative or positive gives you some info. try explore this further at http://www.mathopenref.com/cubicexplorer.html – serg_1 Jul 25 '17 at 10:46
  • Try using the first and second derivative tests, if $f'(x)=0$ then we have a critical point, and since all polynomials are $C^{\infty}$, we have the second derivative also exists and is continuous. If $f''>0$ then we have a minimum, and if $f''<0$ we have a maximum. You might be able to develop an algorithm which uses the first and second derivative tests for overall shape, and some other theorem (maybe IVT?) that tells you exactly where else to plot the function. – Kernel_Dirichlet Jul 27 '17 at 13:15

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We can find the critical points by setting the derivative of the function equal to 0 and solving the equation with the quadratic formula. Because of the square root in the resulting formula, there are 0, 1 or 2 critical points, depending on the sign of $b^2 - 3ac$. $$3ax^2+2bx+c=0 \quad\implies\quad x = \frac{-b \pm \sqrt{b^2-3ac}}{3a}$$

We can find the inflection point by setting the second derivative equal to 0: $$6ax + 2b = 0 \quad\implies\quad x = -\frac{b}{3a}$$

This image shows all the possible shapes:

shapes of cubic function

To find the vertical position of the critical points and the inflection point, plug the found values for $x$ back in the cubic function.

Paul
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  • From the sign of $a$ , what we can get ? – S.H.W Jul 28 '17 at 19:12
  • @S.H.W When $x$ gets big, either positive or negative, $x^3$ gets so big that only the $ax^3$ part of the formula matters when it comes to shape, the value of $bx^2+cx+d$ is relatively small. So for a positive $a$ this means a rapid increase, and for a negative $a$ a rapid decrease, except for the part of the graph that is shown here where $bx^2+cx+d$ does have a significant influence. – Paul Jul 28 '17 at 19:28