Mathematical inquiry processes: Explore, change representations; reason and prove. Conceptual field of inquiry: Differentiation; gradient function; graphs of quadratic and cubic functions.
The prompt follows on from the quadratic function inquiry. Once again it is not possible to satisfy the conditions in the prompt - that is, the gradient of the curve of a quadratic function cannot be the same for two values of x.
The inquiry introduces the derivative of a function with respect to x. In the context of the graph of y = ax2 + bx + c, the derivative is the gradient function. As the notation is arbitrary, the teacher should inform students of its meaning in the orientation phase of the inquiry.
Describing f'(x) as the gradient function immediately suggests a line of inquiry. Students might independently start to wonder about the gradient at x = 1 and x = 2 or be guided to do so. The teacher should provide students with the graph of y = x2 at the start of the inquiry (with an elongated x-axis).
The students might have already met the concept of a tangent to a curve, which they draw by eye and work out the gradient of the straight line. They soon realise that the gradient is different for each value of x and begin to conjecture a relationship between the value of x and the gradient at each point.
Lines of inquiry
1. Student-led exploration
Students use the regulatory cards to decide on the next stage of the inquiry. A common approach is to explore gradients of different parabolas. Students consider points on the graphs of other functions, such as f(x) = 2x2, f(x) = 3x2 and f(x) = 4x2.
They generalise from their results to derive a general formula for the derivative of functions in the form f(x) = ax2, which is f'(x) = 2ax. The teacher chooses a pair of students to explain the generalisation so that the whole class has a model to work towards in further exploration.
The teacher coordinates suggestions for more lines of inquiry:
What happens if the function is in the form f(x) = ax2 + c? Why does the formula remain the same?
What happens if the function is in the form f(x) = ax2 + bx? Why is the gradient function now f('x) = 2ax + b? What is the connection between the change to the formula and the translation of the curve?
For example, the illustration below shows the parabolas of f(x) = x2 and f(x) = x2 + 2x. The second function can be re-written as f(x) = (x + 1)2 - 1 and represents a translation of one unit left and one down. The gradient of the second parabola at x = -2 is equal to the gradient of the first one at x = -1.
What happens to the gradient function for higher-degree polynomials?
2. Teacher-directed derivation
An alternative line of inquiry is to derive the gradient function by considering the chord AB. As B approaches A, the gradient of the chord approximates to the gradient of the tangent at A.
The teacher directs students to complete a table of results for different points on the graph of f(x) = x2. When the coordinates of A are (1,1), the gradient is 2 at the limit (see below).
3. Finding functions for which f'(1) = f'(2)
Another line of inquiry involves an algebraic approach to find a cubic function for which f'(1) = f'(2). See the mathematical notes.