Graphs of functions inquiry

The prompt

Mathematical inquiry processes: Identify and describe properties; extend to other cases. Conceptual field of inquiry: Graphs and equations; gradient; turning point; equations of circles; points of intersection; transformations.

 Jane Moss designed the prompt for her year 12 classes (students aged 17) as a bridge from GCSE to A-level. She mentions the inquiry in her MPhil on inquiry-based tasks

In the form shown in the prompt, the teacher can lead a discussion to draw out the students' existing knowledge and then build upon it. 

Open inquiry

Jane's original task is more open than the prompt. Her initial starting point in classroom inquiry has been:

Take two points (0,8) and (2,4) and investigate.

Jane explains that she would ask students how they can use the points and what they can find out from them. Depending on the students' responses, Jane might provide relevant mathematical terms and use some or all of the questions that follow:

The questions provide a structure if students are struggling. Jane also gives them the set of possible solution curves that are shown in the prompt as further support.

May 2023

Lines of inquiry

Having used the inquiry many times, Jane has compiled a list of lines of inquiry that regularly arise in classrooms:

1. Find the distance and midpoint of the line between the two coordinates and its perpendicular bisector.

2. Find the equations of the straight line and parabola.

3. Deduce the equation of the curve, which can be worked out using simultaneous equations and differentiation.

4. Find the equation of the circle whose centre is (2,4) and circumference passes through (0,8).

The circle could be included in an alternative prompt (see illustration), although the statement is then only partially true. 

Introducing a circle to the inquiry also gives the teacher the opportunity to discuss why a circle is not a function.

5. Find quartic and quintic functions whose graphs pass through the two points.

6. Find a 'family' of cubic equations that pass through the two points and deduce their general form.


7. Work out the points of intersection of the line, curves and circle. 

8. Choose two other coordinates and find functions whose graphs pass through the points.

During their inquiries, Jane encourages students to use Desmos to explore equations, find curves, check conjectures and verify findings.