Compare properties across graphs, tables, rules and descriptions
Compare the same property across different representations · Identify what a finite table does and does not establish · Account for units and domains in comparisons
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Graph features
- Quadratic expressions and linear slopes
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Compare the same property across different representations
- Identify what a finite table does and does not establish
- Account for units and domains in comparisons
Choose one property
A rule, graph, table and verbal description can each represent a function. To compare maxima, first check that each function has a maximum on the specified domain. A quadratic written −(x − h)² + k has unrestricted maximum k at x = h; a graph's peak can be read with its scales.
A table's largest listed value is not necessarily the function's global maximum. If the question guarantees a quadratic and gives enough exact points, derive its rule; otherwise report only what the table establishes.
Compare like with like
Slope from equally spaced table inputs is Δy/Δx, not merely the output difference. Convert units before comparing rates: 0.5 metres per second and 30 metres per minute are equal.
Different domains can change a comparison. A function may reach a peak outside its permitted input interval; checking the unrestricted formula alone would then answer a different question.
Sources for this lesson
Pause and explain
Does a table's largest sampled output always give a function's global maximum?
Worked example
F(x) = −(x − 2)² + 9 on all real x. G is guaranteed quadratic with table (0, 4), (1, 7), (2, 8), (3, 7). Which has the greater maximum?
Show the worked solution
- F has maximum 9 at x = 2.
- G's differences 3, 1, −1 have second difference −2, so its x² coefficient is −1.
- The first three points give G(x) = −x² + 4x + 4 = −(x − 2)² + 8. The fourth checks the rule.
Answer F has maximum 9 and G maximum 8; F's is greater by 1.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
F(x) = 3x + 1. A linear G has points (0, 4), (2, 8), (4, 12). Which increases faster per input unit?
Give me a hint
Divide output changes by input changes.
Compare my reasoning
- F's slope is 3.
- G's slope is (8 − 4)/(2 − 0) = 2, confirmed by the next pair.
F: slope 3, compared with G: slope 2.
Look for these in your work
- Use the input spacing
- Compare the same property
F(x) = −(x − 3)² + 7 on all real x. A graph of G, also on all real x, is a downward parabola with vertex (−1, 9). Compare their maxima and symmetry axes.
Give me a hint
The vertex gives both properties.
Compare my reasoning
- F has maximum 7 and axis x = 3.
- G has maximum 9 and axis x = −1.
G's maximum is 2 greater; axes are x = 3 and x = −1.
Look for these in your work
- Read heights rather than input locations
- State both domains
A table lists H(0) = 2, H(1) = 5, H(2) = 4, with no family or domain stated. May you claim its global maximum is 5? Give a defensible statement instead.
Give me a hint
State only what the observations imply.
Compare my reasoning
- Only three outputs are known. Values at other inputs may be greater.
- 5 is the largest listed output, at input 1. The global maximum is undetermined.
No. The largest sampled value is 5 at x = 1; the global maximum cannot be determined from this information.
Look for these in your work
- Avoid unsupported extrapolation
- Separate sample property from global property
Common mistakes
- Sparse samples do not prove a global peak.
- Compare rates in the same units and on stated domains.
What can you explain now?
Does a table's largest sampled output always give a function's global maximum?
Compare with the explanation
No; values between or beyond the sampled inputs may be larger.
An additional model assumption or full graph information is needed.
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Make it stick
Does a table's largest sampled output always give a function's global maximum?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.