This could be used in a lesson a few days before the main task, or set as a homework. The purpose of the preliminary task is to

remind students of certain ideas or skills related to the main task so that they do not become an artificial barrier in the main task, and

help to inform the way you use the main task by assessing students’ familiarity or confidence with these ideas or skills.

These questions could be used in a mini-whiteboard session

Sketch the graph of a function with two different types of stationary points. Identify the types of stationary points. Shade the portion(s) of the \(x\)-axis where the function is increasing.

Sketch the graph of a function with at least one stationary point. Adapt the sketch on one side of a stationary point to change the type of stationary point. How will this affect the gradient function?

A function has two stationary points. Sketch its gradient function. Adapt this sketch to change the type or number of stationary points. How will this change the original function?

Gradient match is an ‘algebra-free’ resource in which students have to match up graphs of functions with graphs of their derivatives.

Are there some graphs that belong to more than one family?

Follow-up task

We have also suggested options for a follow-up task, which could be used a few days after the main task. This provides an opportunity to revisit key ideas from the main task.

Find the coordinates of each turning point on the graph of \(y=3x^4-16x^3+18x^2\) and determine in each case whether it is a maximum point or a minimum point.

Sketch the graph of \(y=3x^4-16x^3+18x^2\) and state the set of values of \(k\) for which the equation \(3x^4-16x^3+18x^2=k\) has precisely two real roots for \(x\).

In this resource, students are asked to find cubic curves for given stationary point properties. Students tackling this resource should attempt to sketch the cubics with the properties they require before thinking about their equations. (As part of working on this resource, students may consider some parts of the Introductory investigation Curvy cubics which is also at Calculus meets functions.)

We have suggested these preliminary and follow-up tasks as part of a sequence of teaching, but they are intended to be used flexibly. For example, you may prefer to use one of our suggested preliminary tasks as a follow-up task, or vice versa.