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Determine a power-family rule from features and points

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A reciprocal relation has vertical asymptote x=h and horizontal asymptote y=k. Which rule uses these parameters?

\[n=-1,\quad a\ne0,\quad h\ne0\]
Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Forming a power-family rule

Choose the normal form from the family features before finding numerical parameters. The parameters a, h and k control scale, horizontal translation and vertical translation

\[y=\frac{a}{x-h}+k\]
  1. Match the family to its power Reciprocal and square-reciprocal rules use powers -1 and -2, while root and sideways families have different normal forms \(\displaystyle n=-1\Rightarrow y=\frac{a}{x-h}+k,\quad n=-2\Rightarrow y=\frac{a}{\left(x-h\right)^2}+k\)
  2. Read the translations For a reciprocal rule with a nonzero scale a, x=h is the vertical asymptote and y=k is the horizontal asymptote \(\displaystyle y=\frac{a}{x-2}+1\Rightarrow x=2\text{ and }y=1\text{ are the asymptotes}\)
  3. Use a point to find the scale Substitute a point and the translations into the normal form, then solve for a. \(\displaystyle (5,3),\ h=2,\ k=1\Rightarrow a=(3-1)(5-2)=6\)
  4. Verify an unused point Substitute another supplied input into the completed rule and compare its output with the stated point \(\displaystyle y=\frac{6}{x-2}+1,\quad (8,2)\Rightarrow y(8)=\frac{6}{6}+1=2\)

Worked example

A reciprocal relation has vertical asymptote x=3, horizontal asymptote y=-2, and passes through (5,4). Determine its rule and check the value at x=7.

\[x=3,\quad y=-2,\quad (5,4)\]
  1. \(\displaystyle y=\frac{a}{x-3}-2\)Use the reciprocal normal form and the two asymptotes
  2. \(\displaystyle a=(4-(-2))(5-3)=6\times2=12\)Substitute the point to find the scale coefficient
  3. \(\displaystyle y=\frac{12}{x-3}-2\)Insert a=12 into the normal form
  4. \(\displaystyle y(7)=\frac{12}{7-3}-2=3-2=1\)Substitute x=7 to find the requested value

Common mistakes

Using the wrong sign for a required asymptote
✕\(\displaystyle x=4\text{ required}\Rightarrow y=\frac{3}{x+4}+1\)✓\(\displaystyle x=4\text{ required}\Rightarrow y=\frac{3}{x-4}+1\)
Both rules are reciprocal rules, but x+4 is zero at -4 rather than the required 4. Use x-4 to place the vertical asymptote at x=4.
Ignoring a supplied point when choosing the coefficient sign
✕\(\displaystyle (3,-3)\text{ required}\Rightarrow y=\frac{4}{x-2}+1\)✓\(\displaystyle (3,-3)\text{ required}\Rightarrow y=-\frac{4}{x-2}+1\)
With asymptotes x=2 and y=1, the point (3,-3) gives a=(-3-1)(3-2)=-4. A positive coefficient 4 gives y=5 at x=3, not -3.
Replacing the required reciprocal family n=-1 by n=-2
✕\(\displaystyle y=\frac{6}{\left(x-2\right)^2}+1\)✓\(\displaystyle y=\frac{6}{x-2}+1\)
The exponent changes the family, so a square in the denominator is not a reciprocal rule

Difficulty levels

introUse a defining feature and one point to find the scale of a power-family rule
standardForm a translated power-family rule when all defining features are stated
challengeRecover a missing translation from symmetric or related point data
examEnter the parameters and select the formula that matches the family and conditions

Questions about this topic

What do a, h and k represent in a reciprocal rule?+

The nonzero coefficient a controls the scale and orientation, h gives the vertical asymptote x=h, and k gives the horizontal asymptote y=k.

Why does x-h place the asymptote at x=h?+

The denominator is zero when x-h=0, which occurs at x=h.

How can a power-family rule be checked?+

Substitute every supplied point into the completed rule and check that the stated coordinates are obtained

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