A quick way for graphing cubic functions

Question

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 .

Answer 1

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:

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