Derivatives and the Shape of a Graph


In this applet, a quartic (4th degree) polynomial is graphed and we'll explore the relationships between the characteristics of the graph (increasing, decreasing, concave up, concave down) and the characteristics of its derivatives.


  1. See how the positioning sliders work. One shifts the graph of `f` left and right, and the other shifts it up and down. Now check the "Show `f'`" checkbox and see the effect of shifting on the derivative. Is this what you expected? Can you explain the behavior?
  2. Play with the shaping sliders, to see the different ways you can shape `f`. (Incidentally, after appropriate shifting, `f` is a polynomial of the form `f(x) = ax^4 + bx^3 + cx^2 + dx`.)
  3. From the power rule, we know that if `f` is a 4th degree polynomial, then `f'` will be a 3rd degree polynomial and `f''` will be a 2nd degree polynomial. Can you see how this is reflected in the applet?
  4. Check the boxes to show where `f` is increasing and decreasing. Now see how the derivative changes as `f` changes. Specifically, how are the increasing sections of the `f`-graph related to the `f'`-graph? And the decreasing sections?
  5. Check the boxes to show where `f` is concave up and concave down. How are the concave up sections of `f` related to the `f'`-graph?
  6. How are the concave up sections of `f` related to the `f''`-graph?

Check Your Understanding

Click the question marks to reveal the answer.

If `f`... then `f'`... and `f''`...
is positive ? ?
is negative ? ?
is increasing ? ?
is decreasing ? ?
has a max at x = c ? ?
has a min at x = c ? ?
has a terrace point at x = c ? ?
has an inflection point at x = c ? ?