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Converse

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if a^2 + b^2 + c^2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?

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Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Parabolic Patterns

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.

Quadratic Transformations

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem offers a good starting point for exploring relationships between the graphs of linked functions. This task offers learners the chance to practise both manipulating quadratic expressions and working with quadratic graphs.

Possible approach

If you don't have access to a computer room, begin the lesson by asking each learner to select a number, and to perform the following operations, and write down their answer (perhaps on a mini-whiteboard)
Square your number. Add the answer to your number. Subtract 6.  
 
Then ask them to perform the following operations using the same starting number:  
Square your number. Add the answer to your number. Subtract 6. Add 2. 
 
Display a coordinate grid on the board at the front, and choose some students to plot points, the $x$ coordinate being the number they chose, and the $y$ coordinates being their answers - it's a good idea to plot these in different colours on the same graph.  
 
If you have a computer and projector, display the Number Plumber and ask learners to suggest inputs and work out what the outputs will be. 
 
If you do have access to a computer room, begin the lesson by allowing learners to explore the Number Plumber interactivity in pairs. Click here to see how to use the NRICH Number Plumber, or click here to see a video for teachers wishing to create their own examples.
 
Once everyone has had a chance to see what's going on, give them time in pairs to discuss the following questions:
What happens when you input the same number into both functions?
How are the two functions related?
How are the graphs related?
 
Encourage learners to suggest conjectures about relationships between graphs, and perhaps introduce the notation $y=f(x)$ in order to express ideas about the relationship between functions. Software such as the free-to-download Geogebra may be useful for plotting other graphs in order to explore related functions of the form $y=f(x)$ and $y=f(x)+a$, or alternatively learners can be encouraged to work in small groups plotting related graphs on graph paper and sharing their results. As learners begin to notice relationships between the graphs, encourage them to explain why the graphs are linked in this way by relating the differences in the graphs to the differences between the two function machines.
 
After tackling the relationship between $y=f(x)$ and $y=f(x)+a$, the second Number Plumber interactivity offers a chance to explore the relationship between $y=f(x)$ and $y=f(x+a)$ - this can be approached in a very similar way. For those exploring without computers, the two functions are:
$x^2+x-6$ and $(x+2)^2+(x+2)-6$

Key questions

How are the two functions related?
How are their graphs related?
What is the relationship between the graph $y=f(x)$ and the graph $y=f(x)+a$?
What is the relationship between the graph $y=f(x)$ and the graph $y=f(x+a)$?
 

Possible extension

The problem More Quadratic Transformations explores the relationship between $y=f(x)$, $y=af(x)$ and $y=f(ax)$ in a similar way.
 
Learners can apply their understanding of transformation of functions by working on the challenges Parabolic PatternsMore Parabolic Patterns, and Parabolas again.

Possible support

Start by experimenting on what happens to straight line graphs when they are translated in the problem Translating Lines.