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SAMPLE MODELS

Quadratic Functions

Here we present a series of non-standard activities devoted to the main object of school algebra, the quadratic function, and its graph, the parabola.

Transformations of Parabolas

Whereas 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:

  • horizontal and vertical shifts by a unit length,
  • stretches and compressions by a factor of two,
  • reflections in the coordinate axes.

With these restrictions, well-known transformations of the standard parabola y = x2 into parabolas with given coefficients are not always possible, and when they are, they are more difficult, but also more interesting to find.


Playing with the first model, you'll understand better how the restricted transformations work and get to know the interface of the activities.

Activity 1

Use our restricted set of transformations and their combinations to obtain the graphs of quadratic functions listed below from the standard parabola y = x2:

  1. y = –x2 + 2x;
  2. y = 2x2 – 4x + 1;
  3. y = (1/2) * (x – 1/2)2;
  4. y = –x2 + x.

Activity 2

Apply the allowed transformations to the standard parabola y = x2:

  1. to reflect it in the line x = 2;
  2. to reflect it in the line y = 2;
  3. to reflect it in the point (1; 1);
  4. to stretch it by a factor of 2 with respect to point (1; 1).

The current position of the red parabola can be memorized using the blue parabola.

Activity 3

Turn the red parabola into the green one using the given transformations.
The numbered buttons shift tasks.
Hint: the number of each task equals the minimum number of transformations needed to solve it.

Phase Plane of a Quadratic Function

Models 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 = x2 + 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 = p2/4, 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.


Use the model to study the relationship between various properties of parabolas and their representation in the phase plane.

Activity 1

Select the set of all the points in the phase plane (p,q) satisfying the following condition.

The equation x2 + px + q = 0 has:

  1. two positive roots;
  2. two negative roots;
  3. two roots of opposite signs, the root with greater absolute value being positive;
  4. two roots of opposite signs, the root with greater absolute value being negative.

Activity 2

Select the set of all the points in the phase plane (p,q) satisfying the following condition.

The equation x2 + px + q = 0 has:

  1. the root x = 0;
  2. no roots;
  3. a unique root.

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 Functions

Computers 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.


Explore and describe the behavior of the output point as the input point moves along its axis. Do this for the three functions given in the model.

Activity

Find the coefficients A, B, C of a quadratic function y = Ax2 + 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 = ax2 + 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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