This worksheet builds mathematical reasoning, teaching students to form a conjecture about how a parameter affects a function's graph, test it against multiple examples, and generalise the pattern. The worksheet is split into warm-up questions, standard questions and then extension questions to test you.
The concepts it covers:
- A conjecture is an educated guess based on a pattern — it must be tested with multiple examples before being generalised as a rule.
- In y=ax²+c, changing c shifts the graph vertically by exactly c (the y-intercept), while a controls how narrow/wide and which way (up/down) it opens.
- In y=a(x−h)²+k, the vertex is always at the point (h, k) — h shifts horizontally, k shifts vertically.
- In y=mx+c, changing m changes the steepness (gradient); two lines with the same m are parallel and never intersect, regardless of c.
- In y=a×bˣ, changing a scales every y-value by the same factor; increasing the base b (for b>1) makes the function grow faster.
What your child will practise:
- Generating a table of values or set of test points for a family of related functions.
- Comparing outputs across the family to identify a consistent pattern (a conjecture).
- Testing a conjecture against multiple values, not just one, before accepting it as a general rule.
- Stating a general rule in terms of the parameter (e.g. "changing c shifts the graph vertically by c").
- Distinguishing between a genuine pattern and a coincidence by checking enough cases.
Every section opens with a worked example, and the download includes a full answer key with step-by-step solutions and teaching notes on the mistakes students most commonly make.
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