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The prompt
The prompt
Mathematical inquiry processes: Explore, generate examples, and generalise; compare representation. Conceptual field of inquiry: Roots of polynomials; transformation of graphs.
Andrew Blair designed the prompt to launch a unit on the roots of polynomials, although if can also be used if students have already studied the connections between the roots and the coefficients of terms in x.
If the topic is new, the teacher should encourage students to derive the connections from the general equations. Indeed it is possible that the equations (or, at least, the first one below) arise in the question, notice, and wonder phase of the inquiry.
Degree of the polynomial
Degree of the polynomial
Students might infer that the function in the prompt is quadratic because it has two roots. However, the prompt is deliberately ambiguous because repeated roots would make the degree of any polynomial higher than two.
The teacher might specify that the function is quadratic by rewriting the prompt before the inquiry starts or guiding students to that conclusion at the outset. This will narrow the field of inquiry until students are familiar with linear transformations in the context of roots of polynomials.
Question, notice, and wonder
Question, notice, and wonder
Students' initial responses to the prompt have included:
What do 𝞪 and 𝞫 mean?
Is the function quadratic?
You multiply the factors (x - 𝞪)(x - 𝞫) to find the quadratic function.
Can we substitute integers for n, 𝞪 and 𝞫 to check what happens to a function?
We could change the prompt by adding another root. Then the function would be cubic.
The second function looks like a transformation of the first one. We wonder if we could represent it graphically?
Could we make different changes to the roots - add, subtract, and divide by n - and maybe use two or more operations.
Do the two roots have to be transformed in the same way? What happens if we change the roots in different ways? Could we still make a general statement about the change to the function?
Verification
Verification
The generalisation in the prompt is true. Each student might create their own example with different values of 𝞪 and 𝞫 to verify the statement for a particular case (see illustration below). As the inquiry develops, the teacher should encourage students to connect a graphical representation to the linear transformations. Then, when solving problems, students will understand how and why the function changes under different transformations.
For example, if the roots of a quadratic function are transformed from 𝞪 and 𝞫 to 2𝞪 + 1 and 2𝞫 + 1, the parabola is stretched by scale factor two vertical to the y-axis, stretched by scale factor two horizontal to the x-axis, and translated one to the right (see example 7).
A proof of the generaliston in the prompt is shown below. The teacher and students might co-construct the proof to act as a model for attempts to prove generalisations about other linear transformations of roots.
May 2026
Lines of inquiry
Lines of inquiry
Guided inquiry
Guided inquiry
The slides contain lines of inquiry based on the six regulatory cards below. In a guided inquiry, pairs of students suggest how the inquiry could proceed by selecting a card and justifying their selection to the class. For a description of the meaning of each card in the context of the inquiry, see the lines of inquiry below.
The teacher aims to draw out and draw upon students' existing knowledge. If students select this regulatory card, the teacher would co-construct an example (such as example 1).
To decide if the prompt is true, students generate their own examples. They create a polynomial from factors, multiply the roots by a constant, and then attempt to connect the resulting polynomial to the original function (examples 1 and 2).
When enough students have shared their examples, the class might construct a proof collaboratively. This then acts as a model for attempts to prove generalisations about other linear transformations of roots (examples 3 and 4).
As students generate examples, they represent the polynomials graphically and describe the transformations.
Multiplying the roots by a constant gives rise to two stretches - one vertical and the other horizontal. These can be recorded in either order.
However, as the linear transformations become more complex, students identify translations as well. At this stage, it is important to list the order carefully. An intermediary stage can help students identify the transformations more clearly (see below).
Use the 'What-if-not?' strategy from The Art of Problem Posing to change the prompt.
What if the linear transformation is not the multiplication of roots by a constant? Students could use other single operations when they create their own examples (examples 5 and 6).
What if the linear transformation is not one operation? Students use two or three operations to define the new roots (example 7).
What if the function is not quadratic? Students could explore cubic or quartic functions (example 8).
Once students have explored the new roots of a polynomial for a particular case and explained the transformation on a graph, they can make and attempt to prove a general statement (see the examples).
Questions in the A-level Further Mathematics examination specify the polynomial and the linear transformation but not the actual roots. Nevertheless, students can apply the linear transformation to find the new function in terms of the original one (see below and example 9).
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