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The prompt
The prompt
Mathematical inquiry processes: Explore, change representations; reason and prove. Conceptual field of inquiry: Function notation; substitution; graphs of quadratic and cubic functions.
The prompt invites students to explore the contention that a quadratic function can be equal when x is 1, 2, and 3. That this is not possible often becomes clear to students by graphing quadratic functions.
Orientation
The prompt is designed for classes who will recognise the quadratic nature of the function, but would find it a challenge to make meaning of the contention. The orientation phase starts with pairs of students posing questions and making observations in turn:
Could we replace a, b, and c with numbers, including negative numbers?
Is there a specific answer for the values of a, b and c?
What does the f(x) mean?
Does x equal 1, 2, or 3?
Why are f(1), f(2), and f(3) equal?
Once each pair of students has had the opportunity to participate, the teacher aims to construct an understanding of the prompt by probing particular contributions, inviting students to answer questions and, when necessary, explaining the mathematical notation.
Exploration
Through the regulatory cards, students might opt to explore by testing particular cases. The teacher could decide to structure this phase of the inquiry by suggesting quadratic functions to test.
In the shaded part of the table, students notice that it is possible for the function to be equal for two values of x. This often leads to the realisation that if we remove the constraint on the values of x, then there are always pairs of values of x that make the function equal. So, for example in the case of x2 + x +1 (see Graph 1 below), f(-1) = f(0), f(-2) = f(1), and so on.
Graphing
The realisation that a quadratic function can be equal for two values of x leads into a line of inquiry involving cubic functions: Is there a cubic function for which f(1) = f(2) = f(3)?
Using Desmos to graph functions allows students to explore the question. We can see in Graph 2 that there are no two values of x that make the cubic function equal, but it is possible to find three in Graph 3 in the domain between x = -1 and x = approximately 1.65. Students might be encouraged to draw horizontal lines on the graph to find possible solutions, such as f(-0.6) = f(0) = f(1.6) = -1.
However, the approach does not necessarily help us find a cubic function such that for f(x) = ax3 + bx2 + cx + d, f(1) = f(2) = f(3).
To find possible values of a, b, c, and d, the teacher could introduce an algebraic approach (see below in 'Proof').
July 2022
Graph 1
f(x) = x2 + x +1
Graph 2
f(x) = x3 + x2 + x +1 = (x + 1)( x2 +1)
Graph 3
f(x) = x3 - x2 - x -1
Lines of inquiry
Lines of inquiry
1. f(f(1)) = f(f(2)) = f(f(3))
The prompt originated in the Implementing the Mathematical Practice Standards Project to prepare teachers for the Common Core State Standards in the US. It draws on the illustration Making Sense of a Quadratic Function, which also includes the task to determine all the quadratic functions that satisfy f(f(1)) = f(f(2)) = f(f(3)).
2. Proof
The mathematical notes contain a proof that it is not possible to find a, b, and c to satisfy the condition in the prompt. However, students can find values of a, b, and c such that f(1) = f(2), f(1) = f(3), or f(2) = f(3). Similarly, they can deduce the values of a, b, c, and d for cubic functions that satisfy the condition f(1) = f(2) = f(3).
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