Autograph Newsletter 15 – Quadratics

Welcome to the fifteenth Autograph Newsletter! Each jam-packed edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better.
Diagnostic Question
Free Online Autograph Activity
Ideas for Development
Video Tutorials
Handy Autograph Tip
Quadratic graphs and Autograph go together like toast and butter. Indeed, even the most tech-reluctant of teachers may see the benefit of drawing quadratic curves on Autograph. But to leave it at that is a crime against mathematics, because the software can offer so much more. In this newsletter we will look at how to develop the students’ knowledge and understanding of quadratic graphs in a dynamic and interactive way, starting with plotting curves from tables of values and culminating with a beautiful quadratic relationship. I hope you find something useful amongst all of this!
Diagnostic Question
Diagnostic questions are ideal to use at the start of the lesson to enable you to get a quick and accurate picture of your students’ levels of understanding. They are designed in such a way that common misconceptions that your students may hold should steer them to one of the incorrect answers, thus allowing you to learn where the problems lie from their responses. Typically I give my class 30 seconds thinking time and then ask them to hold up their fingers: 1 for A, 2 for B, etc.
Diagnostic Question
Free Online Autograph Activity
Find the Curve
What have these invisible curves got to do with y = x²?
Find the Curve
These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
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
How do you know where it crosses the y-axis?
How do you know where it crosses the x-axis?
How can you work out its minimum point?
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
When your students are ready, test their predictions by turning on  Slow Plot mode and  enter in the relevant equation.
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.
The green curve is y = x² and is used for reference
The red curve is in factorised form, y = (x + a)(x + b)
The blue curve is in completing the square form, y = (x + c)² + d
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
Can students summarise what effect each letter has on the equation and the position of the curve?
Can they explain why this is the case?
Can students change the value of c and d to make the blue curve sit on top of the red curve?
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
The page shows the graph of y = x² +2x – 3, where a = 1, b = 2 and c = -3
How does the solution to the quadratic formula relate to the graph?
Now just calculate the value of the discriminant: b² – 4ac
Use the  constant controller to change the values of a, b and c and calculate the discriminant each time.
Can you make the discriminant negative? What feature does such a graph have?
Can you make the discriminant zero? What feature does the graph have?
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!
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 y-intercept) of the line with the y-axis.
Can you spot a relationship between the y-intercept and the positions of the orange and blue points?
 Drag the orange and blue points to new positions
Can you spot the relationship now?
Use the  Drag button to have a look further up the graph if you need
Tricky: Can you prove this relationship?
Video Tutorials
The following video takes you through, step-by-step, different approaches to graphing quadratics on Autograph.
Handy Autograph Tip
Sometimes on Autograph it is handy to be able to display the full equation of the quadratic curve (or straight line) when changing the constants
Open Autograph in Standard Mode
Enter the equation y = ax² + bx + c
Place three points on the curve
Select these just these three points, right-click and choose Quadratic (3 points) from the menu
Now select just the new curve
Click on text-box and click OK. The equation of the quadratic should be displayed.
Now, if you use the constant controller to change the values of a, b and c you should find that the equation itself automatically updates!

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