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Inverse variation: write the relationship

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Which rule matches inverse variation as the square of q?

\[p \text{ varies inversely as the square of } q\]
Year 11, VCE Units 1 & 2, Chapter 9, 3 difficulty levels, Calculator allowedBuild a worksheet

Writing variation equations

A variation statement names a dependent variable and an independent variable. Direct variation uses multiplication by positive k; inverse variation uses a reciprocal, with the stated power on the independent variable

\[y \propto x^{n} \Rightarrow y = kx^{n},\quad y \propto \frac{1}{x^{n}} \Rightarrow y = \frac{k}{x^{n}}\]
  1. Keep variable roles Keep the dependent variable on the left and apply the stated relationship to the independent variable \(\displaystyle r \text{ varies directly as } s^{2} \Rightarrow r = ks^{2}\)
  2. Use multiplication for direct Direct variation means the dependent variable equals k times the stated power of the independent variable \(\displaystyle p \propto q^{3} \Rightarrow p = kq^{3}\)
  3. Use a reciprocal for inverse Inverse variation means the dependent variable equals k divided by the stated power of the independent variable \(\displaystyle p \propto \frac{1}{q} \Rightarrow p = \frac{k}{q}\)
  4. Put the power on q The named power belongs to the independent variable, including a square-root power \(\displaystyle p \propto \frac{1}{\sqrt{q}} \Rightarrow p = \frac{k}{\sqrt{q}}\)

Worked example

The variable r varies directly as the cube of s. Write an equation using the positive constant k.

\[r \propto s^{3}\]
  1. \(\displaystyle r \text{ is dependent and } s \text{ is independent}\)The variable named first is the dependent variable, and the variable it varies with is independent
  2. \(\displaystyle r \propto s^{3}\)The statement says that r varies directly as the cube of s.
  3. \(\displaystyle r = ks^{3}\)Replace proportionality with multiplication by the positive constant k.

Common mistakes

Direct variation reversed
✕\(\displaystyle r \text{ varies directly as } s^{2} \Rightarrow r = \frac{k}{s^{2}}\)✓\(\displaystyle r \text{ varies directly as } s^{2} \Rightarrow r = ks^{2}\)
Direct variation multiplies by k; the reciprocal is used for inverse variation
Reversing variable roles
✕\(\displaystyle r \text{ varies directly as } s^{3} \Rightarrow s = kr^{3}\)✓\(\displaystyle r \text{ varies directly as } s^{3} \Rightarrow r = ks^{3}\)
The named dependent variable stays on the left, and the stated power applies to the independent variable
Treating inverse as subtraction
✕\(\displaystyle t \text{ varies inversely as } u \Rightarrow t = k - u\)✓\(\displaystyle t \text{ varies inversely as } u \Rightarrow t = \frac{k}{u}\)
Inverse variation is a reciprocal relationship, not a decreasing linear subtraction

Difficulty levels

standardOrdinary and square inverse translation
challengeCube/root translation and cumulative direct/inverse matching
examRecognise inverse first-power or square equations

Questions about this topic

What does k mean in a variation equation?+

k is the positive constant of variation. It is the same for every pair of values in the relationship

How can direct and inverse equations be distinguished?+

A direct equation multiplies the constant by the input raised to the stated power. An inverse equation divides the constant by the input raised to that power

Where does the stated power go?+

The stated power goes on the independent variable, the quantity after the words varies as. The dependent variable remains on the left of the equation

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 9: Variation

In this chapter

Direct variation: write the relationshipDirect variation: find the constant and missing valuesDirect variation: solve a practical problemInverse variation: write the relationshipInverse variation: find the constant and missing valuesInverse variation: solve a practical problem9B Inverse variation: Identify direct or inverse variation from tables and graphs9C Data transformations: Identify direct or inverse variation from tables and graphsSquare and reciprocal data transformationsEvaluate logarithms and antilogarithmsOrders of magnitudeCompare quantities on a log scaleLogarithmic data transformationsFit and use square, reciprocal and logarithmic models