Derivatives & Graphs (DP IB Analysis & Approaches (AA)): Revision Note

Derivatives & Graphs

How are derivatives and graphs connected?

• If the graph of a function is known, or can be sketched, then it is also possible to sketch the graphs of the derivatives and

• The key properties of a graph include

  • the-axis intercept

  • the-axis intercepts – the roots of the function; where

  • stationary points; where

    • turning points – (local) minimum and maximum points

    • (horizontal) points of inflection

  • (non-stationary,) points of inflection

  • asymptotes – vertical and horizontal

  • intervals where the graph is increasing and decreasing

  • intervals where the graph is concave down and concave up

• Not all graphs have all of these properties and not all can be determined without knowing the expression of the function

• However questions will provide enough information to sketch

  • the shape of the graph

  • some of the key properties such as roots or turning points

How do I sketch the graph of y = f'(x) from the graph of y = f(x)?

• The graph of will have its

  • -axis intercepts at the-coordinates of the stationary points of

  • turning points at the-coordinates of the  points of inflection of

• For intervals where is concave up, will be increasing

• For intervals where is concave down , will be decreasing

• For intervals where is increasing, will be positive

• For intervals where is decreasing, will be negative

How do I sketch the graph of y = f''(x) from the graph of y = f(x)?

• First sketch the graph of from, as per the above process

• Then, using the same process, sketch the graph of from the graph of

• There are a couple of things you can deduce about the graph of directly from the graph of

  • The graph of  will have its -axis intercepts at the -coordinates of the points of inflection of 

  • For intervals where is concave up, will be positive

  • For intervals where is concave down, will be negative

Is it possible to sketch the graph of y = f(x) from the graph of a derivative?

• It is possible to sketch a graph of by considering the reverse of the above

  • For intervals where is positive, will be increasing but is not necessarily positive

  • For intervals where is negative, will be decreasing but is not necessarily negative

  • Roots of give the-coordinates of the stationary points of

• There are some properties of the graph of that cannot be determined from the graph of

  • the-axis intercept

  • the intervals for which is positive and negative

  • the roots of

• Unless a specific point the curve passes through is known, the constant of integration cannot be determined

  • the exact location of the curve will remain unknown

  • but it will still be possible to sketch its shape

• If starting from the graph of the second derivative,, it is easier to sketch the graph of first, then sketch