Read intercepts, signs, symmetry and end behaviour
Describe linear, quadratic and reciprocal graph features · Identify extrema and intervals of increase or decrease · Interpret behaviour near excluded inputs and far along a graph
Exam scope and assessment notes
Core and Extended foundations. 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
- Tables and Core function graphs
- Inequalities
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
- Describe linear, quadratic and reciprocal graph features
- Identify extrema and intervals of increase or decrease
- Interpret behaviour near excluded inputs and far along a graph
Read properties from the whole graph
An x-intercept has y = 0; a y-intercept has x = 0, if that input is allowed. Positive regions lie above the x-axis and negative regions below it. Increasing means larger inputs give larger outputs throughout the stated interval; decreasing means the reverse.
For y = −(x − 1)² + 4, the axis of symmetry is x = 1 and the maximum is 4. The roots are −1 and 3; outputs are positive between them, zero at the endpoints, and negative outside. The graph increases to x = 1 and decreases afterwards.
End behaviour and breaks
A line with positive slope increases without an extremum on the whole real domain. An upward parabola eventually rises at both ends; a downward parabola falls at both ends.
For y = 2/x, each branch decreases as x increases within x < 0 or x > 0. This is not decreasing across its entire disconnected domain: compare x = −1 with x = 1. As |x| grows, y approaches zero without reaching it; as x approaches zero from either side, |y| grows without bound. A restricted domain may change extrema and range.
Sources for this lesson
Pause and explain
Is y = 2/x decreasing across every pair in its whole domain?
Worked example
Describe y = −x² + 2x + 3 on all real inputs.
Show the worked solution
- Complete the square: y = −(x − 1)² + 4.
- Roots −1 and 3 give the positive interval −1 < x < 3.
- The graph has maximum 4 at x = 1, symmetry x = 1 and y-intercept 3; both ends fall.
Answer Maximum (1, 4); range y ≤ 4; positive for −1 < x < 3.
Try a topic test
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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.
Find the intercepts, minimum and positive region of y = x² − 4.
Give me a hint
Use y = 0, x = 0 and the square's non-negativity.
Compare my reasoning
- Roots are −2 and 2; the y-intercept is −4.
- Minimum is −4 at x = 0. Outside the roots, x² > 4.
Intercepts (−2, 0), (2, 0), (0, −4); minimum (0, −4); positive for x < −2 or x > 2.
Look for these in your work
- Separate points from intervals
- Exclude roots from strict positivity
Describe the signs, intercepts and far-end behaviour of y = −3/x.
Give me a hint
Zero is excluded from the domain.
Compare my reasoning
- Outputs are positive for negative x and negative for positive x.
- There are no intercepts: x = 0 is forbidden and −3/x cannot equal 0.
- Outputs approach zero as |x| increases.
Positive on x < 0; negative on x > 0; no axis intercepts; approaches y = 0 at both far ends.
Look for these in your work
- Check whether an intercept is allowed
- Do not confuse approach with equality
A model y = −(x − 2)² + 9 is restricted to 0 ≤ x ≤ 1. Find its maximum on this domain; explain why 9 is wrong here.
Give me a hint
The vertex input must belong to the domain.
Compare my reasoning
- The unrestricted vertex is x = 2, outside [0, 1].
- On [0, 1] the graph increases from 5 to 8.
Maximum 8 at x = 1. The unrestricted maximum 9 is not attained on this domain.
Look for these in your work
- Respect the domain
- Check allowed endpoints
Common mistakes
- A reciprocal graph's separate branches must be treated separately.
- An extremum of the unrestricted graph may lie outside a stated domain.
What can you explain now?
Is y = 2/x decreasing across every pair in its whole domain?
Compare with the explanation
No; it decreases on each separate branch only.
At −1 the output is −2, and at 1 it is 2, so the cross-branch comparison increases.
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Make it stick
Is y = 2/x decreasing across every pair in its whole domain?
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.