Take a look at these graphs. The numbered graphs show the gradient functions of ones with letters, but some are missing…

  • Try to pair up a function with its gradient function. Which features of the graphs will you use to justify your reasoning?

  • If a graph can’t be paired up with anything, try to sketch a graph that would complete its pair.

The graphs are available as cards to cut out and sort, including some blank cards for the missing graphs.

Once you have sorted the graphs into pairs of functions \(f(x)\) and \(f'(x)\) and sketched any missing graphs, try to sketch the graphs of the second derivatives.

Also try to sketch graphs of functions whose derivative is \(f(x).\) What do you notice?