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Determine a power-family rule from features and points

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Step 5 of 5

A reciprocal relation must have vertical asymptote x=4 and horizontal asymptote y=1. Which rule matches both requirements?

Year 11, VCE Units 1 & 2, Chapter 6, 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 11, VCE Units 1 & 2, Mathematical Methods, chapter 6: Functions and relations

In this chapter

Classify a relation and state its domain and rangeEvaluate and determine function rulesFind 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 pointsDetermine a circle's equation and featuresAnalyse semicircles and circular regionsApply and describe transformations of a functionIntersections and tangents of a line and a circleEvaluate and interpret a restricted rational modelForm and analyse an inverse relation or function