Simulic
English

Linear, quadratic or exponential? – Tables, differences, ratios and comparing functions

MathematicsFunctions & GraphsAges 15–16

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See a function as a table, a graph and a formula side by side. Constant first differences Δy mean linear, constant second differences Δ²y mean quadratic, and a constant ratio between successive values means exponential. Mystery tables lets students classify random tables (including unequal x-steps and tables that fit no model); Compare puts two functions in different representations – a table, a formula or words – and finds intercepts, rates of change, the break-even point and where an exponential overtakes a linear or quadratic function.

Lesson: Representations of functions; first and second differences; rates of change; comparing linear, quadratic and exponential models

What it shows

A function can be given as a table, a graph or a formula. When the x-values go up in equal steps, a linear function has constant first differences Δy, a quadratic has constant second differences Δ²y, and an exponential function has a constant ratio between successive y-values. Mystery tables trains this test, including tables with unequal x-steps where differences mislead. Compare sets two functions in different representations and finds intercepts, rates of change and intersections. The models are simplified: calls are charged per exact minute and interest is added once a year. See also Linear functions, Graphing quadratics and Exponential function y = aˣ.

How to use

Choose a screen. In Explore, pick a Family, then set Step Δx, Rows and the coefficient sliders; the constant column turns green. In Mystery tables, study the table, tick Show Δx, Δy, Δ²y and ratio columns if you need help, choose Linear, Quadratic, Exponential or None of these, then press Show formula or New table. In Compare two functions, pick a Situation, move the sliders and drag across the graph to read both values.

Parameters you can change

  • Starting screen Explore (table, graph, formula), Mystery tables (classify), Compare two functions
  • Starting family (Explore) Linear y = mx + c, Quadratic y = ax² + bx + c, Exponential y = a · bˣ
  • x-step in the table (Explore) 0.5–2
  • Number of table rows (Explore) 4–8
  • Starting situation (Compare) Two phone plans, Savings: fixed amount or fixed %, k · x² against c · bˣ
  • Show difference and ratio columns from the start (Mystery tables)

Questions to explore

  1. Why can a table with constant Δy fail to be linear when the x-steps are not equal?
  2. For y = ax² + bx + c, how does Δ²y change when the x-step is doubled?
  3. How many years does 10% yearly growth need to overtake adding a fixed amount every year?

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