Direct, inverse or neither
Direct and inverse models have different invariants. Test every supplied pair and interpret the graph together with its domain
- Direct For nonzero inputs, test the ratio of output to input across every supplied pair. The graph is a straight line through the origin \(\displaystyle y=kx\)
- Inverse Test the product of input and output across every pair. For positive quantities and a positive constant, the graph is a decreasing reciprocal curve \(\displaystyle xy=k,\quad x>0\)
- Neither An increasing trend need not have a constant ratio; a decreasing trend need not have a constant product
- 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
- \(\displaystyle l=\frac{48}{w}\)Length is inversely proportional to width because the area is fixed
- \(\displaystyle l=\frac{48}{6}=8\)The length is eight metres. Multiplying the width of six metres by this length gives the fixed area of 48 square metres
Common mistakes
Difficulty levels
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 10 Mathematics, chapter 6: Linear graphs