Evidence for relation families
Classify within the stated families. Check exact equations and use patterns from equally spaced inputs; a finite sample does not prove a unique unrestricted rule
- Linear Constant first differences for equal input steps identify this family. Its graph is a straight line \(\displaystyle y=mx+c\)
- Quadratic Constant non-zero second differences for equal input steps identify this family. Its graph is a parabola \(\displaystyle y=ax^2+bx+c,quad a e0\)
- Circle Points on a circle have a fixed positive distance from its centre. Use equal axis scales. The full circle is not a function of x \(\displaystyle (x-h)^2+(y-k)^2=r^2,quad r>0\)
- Exponential A constant positive non-unit ratio for equal input steps identifies this family. Both growth and decay belong to it \(\displaystyle y=ab^x,quad a>0, b>0, b e1\)
- Avoid shortcuts A curve need not be quadratic. A squared term need not be the highest power. Two squared variables need equal coefficients and the right signs to form a circle
Worked example
For equally spaced inputs 0, 1, 2, 3, classify the pattern among linear, quadratic and exponential families
\[y:3,\ 6,\ 12,\ 24\]
- \(\displaystyle \Delta y:3,\ 6,\ 12\)First differences are not constant
- \(\displaystyle \Delta^2y:3,\ 6\)Second differences are not constant either
- \(\displaystyle 6/3=12/6=24/12=2\)The positive non-unit ratio is constant. Among these families the pattern is exponential growth
Common mistakes
Any curved graph is quadratic
✕\(\displaystyle 2^x=x^2\)✓\(\displaystyle 2^3=8\ne9=3^2\)
Check the position of the variable and the whole patternEvery pair of squares is a circle
✕\(\displaystyle \begin{gathered}x^2+4y^2=16\\\text{is a circle}\end{gathered}\)✓\(\displaystyle \begin{gathered}x^2+y^2=16\\\text{is a circle}\end{gathered}\)
Equal coefficients make the horizontal and vertical radii equal on equally scaled axesA finite table proves a unique rule
✕\(\displaystyle y=2^x\text{ is unique}\)✓\(\displaystyle \begin{gathered}y=2^x\\y=\frac{x^2+x+2}{2}\end{gathered}\)
Both displayed rules pass through (0,1), (1,2) and (2,4). State candidate families and assumptions; finite points alone do not identify an unrestricted ruleDifficulty levels
introIdentify linear and quadratic equations
standardCompare table patterns for three function families
challengeDistinguish circle, quadratic and exponential representations
examClassify all four families from equations, tables and graphs
Questions about this topic
Can exponential graphs decrease?+
Yes. A positive multiplier below 1 gives decay
Must a circle be centred at the origin?+
No. Squared coordinate offsets translate the centre without changing the family
Why are equal input steps important?+
Output differences and ratios must be compared over the same input interval. For a linear rule, use change in y divided by change in x if the steps differ
Is a circle a quadratic relation?+
Its equation has degree two, but a full circle is not a quadratic function y=ax²+bx+c. These questions distinguish the circle from that single-output quadratic family
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs
In this chapter
Expand binomial products and special products10A Expanding and special products: Verify binomial and special-product expansions10B Common factors and difference of squares: Factorise common factors and differences of squares10B Common factors and difference of squares: Verify common-factor and square-difference products10C Quadratic factorisation and fractions: Factorise a quadratic expression completelySimplify an algebraic fraction by factorising and cancelling10C Quadratic factorisation and fractions: Verify monic and simple non-monic factorisationsVerify quadratic fraction cancellations and exact valuesSolve a quadratic equation by taking square rootsSolve a quadratic equation using the null factor lawSet up and solve a quadratic application problemIdentify and interpret parabola features10G Parabola features and transformations: Distinguish linear and quadratic relationsRead and use circle equations and graphsRead and use exponential growth and decay graphs10H Circles and exponential graphs: Classify a relation as linear, quadratic, circle or exponentialConnect reciprocal rules, tables and graphs