Linear and quadratic evidence
A table supplies a pattern over its listed points. A full rule can establish the relation for its stated domain
- Simplify first A squared term must remain with a nonzero coefficient for a polynomial rule to be quadratic
- First differences Equal input steps and constant first differences are consistent with a linear pattern
- Second differences Equal input steps and constant nonzero second differences are consistent with a quadratic pattern
- Read graphs carefully A line has constant gradient. A parabola is smooth and symmetric about a vertical axis, with changing gradient
- Respect the evidence A finite sample cannot uniquely determine all possible functions beyond those points. Use the families and assumptions stated in the question
Worked example
Classify the pattern with inputs 0,1,2,3 and outputs 2,5,12,23.
\[x:0,1,2,3\]
- \(\displaystyle \Delta y:3,7,11\)Subtract each output from the next
- \(\displaystyle \Delta^2y:4,4\)The first differences increase by 4 each time
- \(\displaystyle y=2x^2+x+2\)A quadratic rule fits all listed values. This is a consistent model, not a proof that no other function could fit them
Common mistakes
Calling every curved graph quadratic
✕\(\displaystyle \text{curved}\Rightarrow\text{quadratic}\)✓\(\displaystyle \text{check the defining pattern}\)
Many relation families are curvedIgnoring input spacing
✕\(\displaystyle \begin{gathered}\text{unequal differences}\\\Rightarrow\text{quadratic}\end{gathered}\)✓\(\displaystyle \text{check equal input steps first}\)
Unequal input intervals can give unequal differences even for a lineClassifying before cancellation
✕\(\displaystyle x^2-x^2+x\text{ is quadratic}\)✓\(\displaystyle x^2-x^2+x=x\)
The highest nonzero power after simplification determines polynomial degreeDifficulty levels
introClassify simplified linear and quadratic equations
standardRecognise linear and quadratic tables
challengeIdentify quadratic equations, tables and graphs with evidence
examDistinguish both families across equations, tables and graphs
Questions about this topic
Are changing first differences enough for a quadratic?+
No. Check that second differences are constant and nonzero for equally spaced inputs
What if the inputs are not equally spaced?+
For a possible line compare change in output divided by change in input. Do not use equal-step difference rules without checking the input spacing
Why can an exponential also have changing differences?+
An exponential is multiplicative: equal input steps give a constant ratio when the relevant outputs are nonzero
Does a zero second difference mean quadratic?+
No. Constant first differences give zero second differences; the pattern can be linear
Can a graph window make a curve look almost straight?+
Yes. Check the algebra or numerical changes and use an appropriate scale instead of appearance alone
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