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Identify direct or inverse variation from tables and graphs

Start a set of 8
Step 6 of 19
Pairs (1,9),(2,7),(3,5). Row 1: x 1, y 9. Row 2: x 2, y 7. Row 3: x 3, y 5xy192735

The output decreases. Does that make this table inverse proportion?

Year 11, VCE Units 1 & 2, Chapter 9, 4 difficulty levels, Calculator allowedBuild a worksheet

Direct, inverse or neither

Direct and inverse models have different invariants. Test every supplied pair and interpret the graph together with its domain

\[\begin{aligned}y&=kx\\y&=\frac{k}{x}\quad(x\ne0)\end{aligned}\]
  1. Direct For nonzero x, test y/x across every supplied pair. The graph of y=kx is a straight line through the origin \(\displaystyle y=kx\)
  2. Inverse Test xy across every pair. For positive quantities and positive k, y=k/x is a decreasing reciprocal curve \(\displaystyle xy=k,\quad x>0\)
  3. Neither An increasing trend need not have a constant ratio; a decreasing trend need not have a constant product
  4. Limit the conclusion Matching finite data supports the candidate model for those observations. Check assumptions before extending it

Worked example

A rectangular panel has a fixed area of 48 m². Its width is w metres and length is l metres. Explain the model and predict the length at width six metres

\[wl=48,\quad w>0\]
  1. \(\displaystyle l=\frac{48}{w}\)Length is inversely proportional to width because the area is fixed
  2. \(\displaystyle l=\frac{48}{6}=8,\quad 6\times8=48\)The length is eight metres

Common mistakes

Using the trend alone
✕\(\displaystyle y=12-2x\text{ is inverse}\)✓\(\displaystyle y=\frac{24}{x}\text{ is inverse}\)
A falling line and a reciprocal curve are different. Check the product
Accepting only two matching pairs
✕\(\displaystyle 4/1=8/2\Rightarrow\text{all pairs fit}\)✓\(\displaystyle 15/3=5\ne4\)
The third pair can contradict the proposed constant ratio

Difficulty levels

introClassify direct and inverse tables
standardClassify direct, inverse and neither graphs and calculate the supporting invariant
challengeCheck ordinary or explicitly stated square invariants; classify neither cases
examDistinguish the supplied table or graph relationship

Questions about this topic

Is every decreasing graph inverse proportion?+

No. A constant product is required. A decreasing straight line usually does not have that property

Must inverse proportion always be positive?+

No in pure algebra, but the physical quantities studied here use positive inputs and a positive constant. Other domains need their own interpretation

Do a few data points prove a model?+

They can support or disprove a candidate model for the displayed data. They do not prove it applies to all unobserved inputs

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