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Contents 1. Compass & Straightedge 2. Triangle: Points & Lines 3. Economical Constructions 4. 3D Workshop 5. Wire Puzzles 6. Quadratic Functions 7. Conics and Other Curves 8. Linkages 
Quadratic FunctionsHere we present a series of nonstandard activities devoted to the main object of school algebra, the quadratic function, and its graph, the parabola. Transformations of ParabolasWhereas transformations of function graphs, in particular, of parabolas, are quite a standard topic in the school curriculum, we give them a new aspect by restricting the "magnitudes" of transformations. Namely, in the following activities one is allowed to perform only these transformations:
With these restrictions, wellknown transformations of the standard parabola y = x^{2} into parabolas with given coefficients are not always possible, and when they are, they are more difficult, but also more interesting to find.
Activity 1Use our restricted set of transformations and their combinations to obtain the graphs of quadratic functions listed below from the standard parabola y = x^{2}:
Activity 2Apply the allowed transformations to the standard parabola y = x^{2}:
The current position of the red parabola can be memorized using the blue parabola. Activity 3Turn the red parabola into the green one using the given transformations.
Phase Plane of a Quadratic FunctionModels in this section contain two coordinate frames. One of them is the (x,y)plane with the parabola given by the equation of the form y = x^{2} + px + q. The other one, called the phase plane of this quadratic function, is the (p,q)plane with the point (p,q) in it. This point represents the quadratic function above; by dragging it in the plane, we change the coefficients of the function and move the parabola. Parabolas satisfying a certain property are represented by some set or points in the phase plane. For example, the parabola q = p^{2}/4, called the discriminant parabola, divides the phase plane into two areas: points in the area under the parabola correspond to quadratic functions that have zeros and the area above the parabola represents functions that have no zeros.
Activity 1Select the set of all the points in the phase plane (p,q) satisfying the following condition. The equation x^{2} + px + q = 0 has:
Activity 2Select the set of all the points in the phase plane (p,q) satisfying the following condition. The equation x^{2} + px + q = 0 has:
If some line, such as a ray or a piece of parabola, belongs to the area in question, it must also be selected. Dynagraphs of Quadratic FunctionsComputers allow to implement the method of function visualization different from ordinary graphs, which is known as dynagraphs. We take two parallel number axes and join a point x on one of them, the input axis, to the point y = f(x) on the other, the output axis.
ActivityFind the coefficients A, B, C of a quadratic function y = Ax^{2} + Bx + C given its dynagraph, if it is known that they are integers ranging from –5 to 5. To determine A, B, C, you can assign to x numeric values of your choice using the slider x and the button Go to and observe the values taken by y. To check the answer, you are given the dynagraph of a second function y = ax^{2} + bx + c with coefficients a, b, c specified by sliders. Set the values of a, b, and c equal to the A, B, and C you've found and check if the two dynagraphs work identically using the button "Check yourself."

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