Reciprocal family features
A reciprocal family is described by its power, two asymptotes and signed scale. The asymptotes restrict the domain and range, while the power and scale determine the branch orientation
\[f(x)=\frac{a}{\left(x-h\right)^p}+k,\quad p\in\{1,2\},\quad a\ne 0\]
- Find the vertical asymptote Set the denominator base to zero. The solution is the excluded input \(\displaystyle x-h=0\Rightarrow x=h\)
- Read the horizontal asymptote The term outside the fraction gives the excluded output value \(\displaystyle f(x)=\frac{a}{\left(x-h\right)^p}+k\Rightarrow y=k\)
- Exclude the asymptote values The vertical asymptote is excluded from the domain. The horizontal asymptote is excluded from the range \(\displaystyle D=\mathbb{R}\setminus\left\{h\right\},\quad R=\mathbb{R}\setminus\left\{k\right\}\text{ when }p=1\)
- Use power parity and scale sign An odd reciprocal power gives opposite diagonal branches, while an even power puts both branches on the side set by the sign of the scale \(\displaystyle \frac{3}{x-1}+2\text{ has opposite branches};\quad\frac{3}{\left(x-1\right)^2}+2>2\)
Worked example
For the function shown, with real inputs in its maximal domain, state its asymptotes, domain and range, and find its value at x=5
\[f(x)=\frac{16}{\left(x-1\right)^2}+2\]
- \(\displaystyle x-1=0\Rightarrow x=1\)Set the denominator base to zero to find the vertical asymptote
- \(\displaystyle y=2\)The constant outside the fraction gives the horizontal asymptote
- \(\displaystyle D=\mathbb{R}\setminus\left\{1\right\},\quad R=\left(2,\infty\right)\)The input x=1 is excluded, and the positive square-reciprocal term keeps every output above 2
- \(\displaystyle f(5)=\frac{16}{\left(5-1\right)^2}+2=\frac{16}{16}+2=3\)At x=5 the denominator is 16, so the function value is 3
Common mistakes
Translation sign in the denominator
✕\(\displaystyle g(x)=\frac{6}{x+4}-1\Rightarrow x=4\)✓\(\displaystyle g(x)=\frac{6}{x+4}-1\Rightarrow x=-4\)
The denominator is zero at x=-4, so that is the vertical asymptoteAn asymptote is included
✕\(\displaystyle f(x)=\frac{3}{x-2}+1\Rightarrow D=\mathbb{R}\)✓\(\displaystyle f(x)=\frac{3}{x-2}+1\Rightarrow D=\mathbb{R}\setminus\left\{2\right\}\)
A vertical asymptote is approached but cannot be an input in the domainSquare-reciprocal branches are treated as reciprocal branches
✕\(\displaystyle f(x)=\frac{4}{\left(x-1\right)^2}+2\text{ has branches on opposite sides of }y=2\)✓\(\displaystyle f(x)=\frac{4}{\left(x-1\right)^2}+2>2\)
The squared denominator is positive on both sides, so a positive scale puts both branches above the horizontal asymptoteDifficulty levels
introIdentify the base reciprocal or square-reciprocal family and its excluded zero value
standardFind asymptotes, restrictions and orientation after one translation and a signed scale
challengeAnalyse both translations from an unnormalised denominator and recognise the rule from its features
examSelect a feature set or formula that distinguishes the two reciprocal powers and coefficient signs
Questions about this topic
Why is the vertical asymptote excluded from the domain?+
At the vertical asymptote the denominator is zero, so the function is not defined there
How does the power change the branch shape?+
For power 1 the branches lie in opposite diagonal regions. For power 2 the two branches lie on the same side of the horizontal asymptote
How does the sign of a affect a square-reciprocal function?+
If a is positive, both branches are above y=k. If a is negative, both branches are below y=k
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