Transformed sine and cosine rules
Use the canonical form with a positive inner frequency so each parameter has one role. The outer coefficient controls vertical scale and reflection, while n and h control horizontal features
\[f(x)=aF\left(n(x-h)\right)+k,\quad T=\frac{2\pi}{n}\]
- Read the outer coefficient The amplitude is the positive magnitude of the outer coefficient, and a negative coefficient reflects the graph \(\displaystyle a=-3\Rightarrow A=|a|=3\)
- Find the period For sine and cosine, divide the base period 2π by the positive inner frequency \(\displaystyle T=\frac{2\pi}{n}=\frac{2\pi}{2}=\pi\)
- Read the phase shift The factor x−h shifts the graph right by h. \(\displaystyle 2\left(x-\frac{\pi}{4}\right)\Rightarrow\text{shift right by }\frac{\pi}{4}\)
- Find the range The range is centred at k and extends A units in both directions \(\displaystyle k=2,\ A=3\Rightarrow-1\le f(x)\le5\)
Worked example
Use the exact features to form the canonical sine rule. It crosses its mean upwards at the stated input
\[M=5,\quad m=-1,\quad T=\pi,\quad (x,y)=\left(\frac{\pi}{4},2\right)\]
- \(\displaystyle A=\frac{5-(-1)}{2}=3\)The amplitude is half the distance between the maximum and minimum
- \(\displaystyle k=\frac{5+(-1)}{2}=2\)The mean is the midpoint of the extrema
- \(\displaystyle T=\pi\Rightarrow\frac{T}{\pi}=1\Rightarrow n=\frac{2}{1}=2\)The sine base period 2π is divided by n to give the stated period
- \(\displaystyle \frac{h}{\pi}=\frac14,\quad a=A=3\Rightarrow f(x)=3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)+2\)A positive sine crosses its mean upwards at its phase shift
Common mistakes
Confusing amplitude and period
✕\(\displaystyle A=2,\quad\frac{T}{\pi}=3\)✓\(\displaystyle A=3,\quad\frac{T}{\pi}=1\)
The outer coefficient sets the amplitude and the inner frequency sets the period, so the two roles must not be exchangedReversing the phase shift
✕\(\displaystyle 2\left(x-\frac{\pi}{4}\right)\Rightarrow\text{shift left by }\frac{\pi}{4}\)✓\(\displaystyle 2\left(x-\frac{\pi}{4}\right)\Rightarrow\text{shift right by }\frac{\pi}{4}\)
Subtracting the phase shift inside the function moves the graph rightIgnoring the mean in the range
✕\(\displaystyle f(x)=-3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)+2\Rightarrow-3\le f(x)\le3\)✓\(\displaystyle f(x)=-3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)+2\Rightarrow-1\le f(x)\le5\)
The range is centred at the vertical translation k, so the amplitude is added to and subtracted from k.Difficulty levels
introIdentify one parameter or one feature of a transformed sine or cosine rule
standardFind the main transformations and key points of a rule with several parameters
challengeReconstruct a canonical rule from its period, range and phase information
examCompare or match transformed rules using their algebraic and graphical features
Questions about this topic
Why is amplitude the absolute value of the outer coefficient?+
Amplitude is a positive distance from the mean line. A negative outer coefficient reflects the graph but does not make that distance negative
How is the period found?+
For sine and cosine, divide the base period by the positive inner frequency to find the period
How does the range change after a vertical translation?+
The amplitude gives the distance from the mean to each endpoint of the range
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