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The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

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What's That Graph?

Stage: 4 Challenge Level: Challenge Level:1

This problem also appears on Underground Mathematics, where you can find more materials to support classroom use. 

Why do this problem?

This problem offers students the chance to explore functions and graphs in real-life contexts. There is an opportunity for students to use their scientific understanding as they are invited to suggest plausible processes for each graph.

Possible approach

Hand out this worksheet with copies of the eight graphs, and ask students to discuss in pairs what physical processes they could represent. Make it clear that there are a lot of possible answers:
"I want you to be prepared to justify that the processes you suggest correspond to the graphs you have been given".

After students have had time to come up with processes for each graph, invite them to share their suggested processes and justifications. Encourage the class to be critical of the suggestions.
Next, reveal that the graphs were actually generated from the processes listed on this worksheet. Hand out the worksheet, and ask students to work in pairs to match the processes (and equations if appropriate) to the graphs.

To ensure that students think critically about matching the processes to the graphs, students could be required to present their solutions on a poster with each graph and process (and equation if appropriate) accompanied by a short sentence explaining WHY they match.  

Key questions

What are the key features of each graph?  
For each process, roughly what shape graph would you expect? Which graphs or equations might be consistent with this?

Where might the axes be placed on the graphs? What scale might you put on the axes?

Possible extension

Ask students to suggest realistic values for the constants A, B and C in the equations of the graphs.
Whose Line Graph Is it Anyway? is a similar but more challenging problem using functions met at Stage 5.

Possible support

Start by giving students the processes and ask them to discuss and sketch what the graphs might look like first. Clarify the appropriate labels for each of the axes.