ECON1003 · Quantitative Methods In Economics
Non-Linear Functions
When the line bends, three function families do the work in ECON1003 — and one big idea links them. Quadratics solve with the formula, and the discriminant Δ = b² − 4ac tells you how many real roots exist before you compute anything; their parabola has a turning point you must locate. Exponentials are the natural engine of growth, with e at the centre and the index rules to wield. Logarithms are the inverse that undoes an exponential, letting you solve for an unknown power. The formula sheet hands you the quadratic formula, but the log and index rules you must use fluently. This chapter drills each family on economic shapes — revenue parabolas, compound growth, breaking a price out of an exponential — and is core midterm material.
What this chapter covers
- 012.1 Quadratics: the formula and the discriminant
- 02Locating the turning point of a parabola
- 032.2 Transformations of functions
- 042.3 Exponential functions and the number e
- 052.4 Logarithms and the log/index rules
- 062.5 Exponential growth and decay
Worked example: solve a quadratic and read its parabola
- +1Identify a, b, c. a = 2, b = −7, c = −9.
- +1Discriminant first. Δ = (−7)² − 4(2)(−9) = 49 + 72 = 121 > 0 ⇒ two real roots (√121 = 11).
- +1Apply the quadratic formula. x = (7 ± 11)/4 ⇒ x = 4.5 or x = −1.
- +1Axis of symmetry. Midway between the roots: x = (4.5 + (−1))/2 = 1.75.
- +1Turning-point y and type. y = 2(1.75)² − 7(1.75) − 9 = −15.125; since a > 0 it is a minimum.
Key terms
- Discriminant
- The quantity Δ = b² − 4ac under the root in the quadratic formula. It tells you the number of real roots before you compute: Δ > 0 gives two, Δ = 0 gives one, Δ < 0 gives none.
- Turning point
- The vertex of a parabola, sitting on the axis of symmetry midway between the roots. It is a minimum when a > 0 (the parabola opens up) and a maximum when a < 0.
- The number e
- The base of natural exponential growth, e ≈ 2.718. Continuous compounding, natural growth and decay all use eˣ; its derivative and integral are themselves, which is why it sits at the centre of the calculus topics.
- Logarithm
- The inverse of an exponential: logₐx answers 'to what power must a be raised to give x?'. It is the tool that solves for an unknown power, undoing an exponential. The natural log ln uses base e.
- Exponential growth
- A quantity that multiplies by a constant factor each period, modelled y = A·eˣᵗ (or A(1 + r)ᵗ). Unlike linear growth it accelerates; logs are how you solve such a model for the time or rate.
Non-Linear Functions FAQ
Why should I compute the discriminant before solving?
Because Δ = b² − 4ac tells you how many real roots exist before any arithmetic. If Δ < 0 there is no real root — write 'no real solution' rather than forcing a number. Computing Δ first also catches the common sign slip, where a negative b is mistakenly left negative after squaring (it becomes positive).
When do I need logs rather than just exponentials?
Whenever the unknown is in the exponent. If a model says A(1 + r)ᵗ = target and you must find t, you take logs of both sides to bring t down. Exponentials build the growth; logs solve it for the power. The log and index rules are not on the formula sheet, so practise them until fluent.
How do I find a parabola's maximum or minimum?
The turning point sits on the axis of symmetry, midway between the two roots (or at x = −b/2a). Substitute that x back to get the y. The sign of a decides the type: a > 0 opens upward so the turning point is a minimum; a < 0 opens downward so it is a maximum. This is the algebraic route; differentiation gives the same answer.
Exam move
Make the discriminant a reflex first move on any quadratic — it tells you the number of roots and guards against the sign slip inside the root. For the turning point, remember the axis of symmetry sits midway between the roots and the sign of a sets max vs min. Drill the index and log rules until automatic, since they are NOT on the formula sheet and every exponential-growth or 'solve for the unknown power' question depends on them. Tie each family to its economic shape (revenue parabola, compound growth, decay) so you recognise on sight which tool a worded question wants.
