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Select and interpret a function model

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What evidence should be checked before selecting a function model?

Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Selecting and interpreting models

A model is selected from the qualitative pattern of the data, such as constant change, turning behaviour, repetition or approach to a level. The variables, units and meaningful domain must be stated before an output is interpreted

\[y=f(x),\quad x\in\mathcal{D}_{\text{model}}\]
  1. Check the complete pattern A single point can fit many families. Compare differences or ratios at equally spaced inputs, and check turning behaviour, repetition or approach to a level \(\displaystyle x:0,1,2,3;\quad y:4,8,12,16\quad\Rightarrow\quad\Delta y=4\)
  2. State the meaningful domain A contextual model is used only for inputs represented or justified by the situation \(\displaystyle 0\le t\le12\)
  3. Evaluate inside the domain Substitute the input and complete the exact calculation before interpreting the output \(\displaystyle y(3)=5+2(3)=11\)
  4. Interpret the final output A limiting value is approached rather than attained, and a count is rounded once at the final contextual step \(\displaystyle y=40-8\left(\frac{1}{2}\right)^t\to40,\quad y\ne40\text{ for finite }t\)

Worked example

The number of stored records is modelled by the exact rule below. Find the modelled number after 3 batches

\[R(n)=20+5\left(2\right)^n,\quad 0\le n\le6,\quad R(3)=?\]
  1. \(\displaystyle 0\le3\le6\)The requested batch number is inside the meaningful domain
  2. \(\displaystyle 2^3=8\)Evaluate the changing factor for batch 3
  3. \(\displaystyle 5\times8=40\)Multiply the changing factor by its coefficient
  4. \(\displaystyle R(3)=20+40=60\)Add the baseline to obtain 60 stored records

Common mistakes

Choosing from one data point
✕\(\displaystyle (0,12)\Rightarrow\text{the model is linear}\)✓\(\displaystyle x:0,1,2,3;\quad y:12,16,20,24\quad\Rightarrow\quad\Delta y=4,4,4\quad\Rightarrow\quad\text{the pattern is linear}\)
One data point can fit many families, so the complete pattern must be checked before a family is selected. Constant-difference and constant-ratio tests require equally spaced inputs
Using additive differences for exponential change
✕\(\displaystyle x:0,1,2,3;\quad y:6,12,24,48\quad\Rightarrow\quad\Delta y=6\quad\Rightarrow\quad\text{constant additive change}\)✓\(\displaystyle x:0,1,2,3;\quad y:6,12,24,48\quad\Rightarrow\quad\frac{12}{6}=\frac{24}{12}=\frac{48}{24}=2\quad\Rightarrow\quad\text{constant ratio}\)
An exponential pattern is identified by a constant ratio after allowing for any baseline, not by a constant additive difference
Treating a limiting value as attained
✕\(\displaystyle y=30-6\left(\frac{1}{2}\right)^t\quad\Rightarrow\quad y=30\text{ at a finite }t\)✓\(\displaystyle y=30-6\left(\frac{1}{2}\right)^t\quad\Rightarrow\quad y\to30\text{ as }t\to\infty,\quad y\ne30\text{ for finite }t\)
A positive exponential term remains for every finite t, so the model approaches 30 without reaching it

Difficulty levels

introRecognise a function family from a qualitative pattern in a table or feature summary
standardEvaluate an exact model at an admissible input and interpret one feature
challengeApply the meaningful domain, interpret limiting or periodic behaviour and reject an out-of-domain value
examCompare two plausible models and select the claim supported at an in-domain input

Questions about this topic

How can a table suggest a linear model?+

If equally spaced inputs give the same additive change in the outputs, the table has a linear pattern over that interval

How are exponential growth and decay recognised?+

After allowing for a baseline when necessary, equal input steps give a constant multiplicative ratio in the changing part

Why is a model's meaningful domain important?+

A formula can produce an algebraic value outside the situation studied, but that value is an extrapolation and may not describe the real context

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs

In this chapter

Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model