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.

One way to approach this is to group the graphs with letters into different “families” and match these groups with gradient function graphs with appropriate features. Can you explain why this approach could lead to the following groups?

  • Graphs A and E are both continuous and their gradients appear to vary smoothly.

  • Graphs D and F have turning points so their gradient function graphs must meet the \(x\)-axis. Or we could say that \(D\) and \(F\) have both positive and negative gradients.

  • Graphs B and C have no turning points. Their gradients are either always positive or always negative.

Now think about how to pair these cards up within the groups.

Aligning cards vertically can help with spotting related features. What feature of a gradient function will need to be aligned with

  • … a stationary point?

  • … an asymptote?

What feature will a function exhibit if the gradient function

  • … is above the horizontal axis?

  • … cuts the horizontal axis?

  • … has a stationary point?