Ideas for Development 
The following ideas for developing the topic of quadratic graphs require the full version of Autograph. 

Activity 1 – Sketching Quadratics 
Download 1. Sketching Quadratics.agg 


This file can be used to encourage students to spot the key features of quadratic curves to enable them to make a good sketch.
Challenge: Sketch y = x² – 4 

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How do you know where it crosses the yaxis? 
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How do you know where it crosses the xaxis? 
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How can you work out its minimum point? 
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Can you figure out some other points? For example, when x = 1, y = ? 


Any time a student finds a point, get them to come up and either use the Scribble tool to mark it, or drag one of the circled points into position 

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When your students are ready, test their predictions by turning on Slow Plot mode and enter in the relevant equation. 
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You can then use this same approach to analyse the key features of any quadratic graph 

Activity 2 – Different Forms of Quadratic Graphs (factorised and completing the square) 
Download 2. Different Forms of Quadratics.agg 


This file can be used to investigate how the different forms of quadratic expressions are represented graphically. 

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The green curve is y = x² and is used for reference 
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The red curve is in factorised form, y = (x + a)(x + b) 
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The blue curve is in completing the square form, y = (x + c)² + d 
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Students can use the constant controller to investigate what happens to the equation of the curves and their position when the values of constants a, b, c and d are altered 
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Can students summarise what effect each letter has on the equation and the position of the curve? 
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Can they explain why this is the case? 
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Can students change the value of c and d to make the blue curve sit on top of the red curve? 
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Can students change the value of a and b to make the red curve sit on top of the blue curve? Why not? 

Activity 3 – The Discriminant 
Download 3. The Discriminant.agg 


We can use Autograph to investigate how the value of the discriminant in the quadratic formula effects the position of the graph of quadratic equation 

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The page shows the graph of y = x² +2x – 3, where a = 1, b = 2 and c = 3 
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How does the solution to the quadratic formula relate to the graph? 
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Now just calculate the value of the discriminant: b² – 4ac 
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Use the constant controller to change the values of a, b and c and calculate the discriminant each time. 
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Can you make the discriminant negative? What feature does such a graph have? 
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Can you make the discriminant zero? What feature does the graph have? 
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Can you explain why this is the case? 

Activity 4 – A Quadratic Relationship 
Download 4. A Quadratic Relationship.agg 


The idea for this activity came from Mike Wakeford who found it in an old geometry book! 

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The page on the left shows the curve y = x². A straight line has then been constructed through the following two points:
1. The intersection of the vertical line through the orange point
2. The intersection of the vertical line through the green point, which has been created using the negative of the blue vector!
Finally, I have marked the intersection (the yintercept) of the line with the yaxis. 
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Can you spot a relationship between the yintercept and the positions of the orange and blue points? 
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Drag the orange and blue points to new positions 
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Can you spot the relationship now? 
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Use the Drag button to have a look further up the graph if you need 
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Tricky: Can you prove this relationship? 
