Cubic and rational function analysis – derivatives, extrema, asymptotes
MathematicsFunctions & GraphsAges 17–18
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Functions & Graphs
Plots y = ax³ + bx² + cx + d or y = (ax + b)/(cx + d) alongside a variation table built live from the sign of y'. Drag point M along the curve to see the tangent line, the derivative's sign, and the interval of monotonicity being examined; change the coefficients to make extrema appear or vanish, or to shift the vertical and horizontal asymptotes.
Lesson: Monotonicity and extrema, asymptotes, curve sketching
What it shows
The sign of the derivative decides the shape of a graph. For the cubic y = ax³ + bx² + cx + d, y′ = 3ax² + 2bx + c: when b² − 3ac > 0 it changes sign twice, giving a local maximum and a local minimum; otherwise the function is monotonic with no extrema. The inflection point at x = −b/(3a) is the centre of symmetry. For y = (ax + b)/(cx + d), y′ = (ad − bc)/(cx + d)² never changes sign, so there are no extrema, only the asymptotes x = −d/c and y = a/c.
How to use
Choose Cubic or Rational, then change Coefficient a, b, c and d one at a time while the variation table rebuilds. Drag point M along the curve: the tangent slopes upward where y′ > 0, downward where y′ < 0, and is horizontal at each extremum. With the cubic, ask students to find coefficients that make the two extrema merge and vanish; with the rational function, watch both asymptotes move as c and d change.
Parameters you can change
- Function type y = ax³ + bx² + cx + d, y = (ax + b)/(cx + d)
- Coefficient a -3–3
- Coefficient b -6–6
- Coefficient c -6–6
- Coefficient d -6–6
Questions to explore
- Which condition on a, b and c gives a cubic two extrema, and can you confirm it by adjusting the coefficients?
- If y′(x₀) = 0, must x₀ be a maximum or minimum, or can you find a cubic where it is not?
- Why does y = (ax + b)/(cx + d) never have a local maximum or minimum?
