Draw linear, quadratic and reciprocal graphs from tables
Construct accurate tables for Core function families · Plot curves with suitable scales and domain breaks · Read approximate solutions from intersections
Section Paper 1 · Historical Core · non-calculator
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
- Function notation
- Cartesian coordinates and signed arithmetic
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
- Construct accurate tables for Core function families
- Plot curves with suitable scales and domain breaks
- Read approximate solutions from intersections
A graph records solutions
Each point (x, y) on a graph satisfies its equation. Choose input values across the required interval, evaluate them carefully and label both axes with consistent scales. A straight line needs aligned points; a parabola needs a smooth curve and enough values around its turning point.
Core families include y = ax + b, y = x² + ax + b, y = −x² + ax + b and y = a/x with integer constants. Parentheses matter when squaring a negative input: (−2)² = 4, while −(−2)² = −4.
Reciprocal branches and intersections
For non-zero a, y = a/x is undefined at x = 0. Plot negative and positive inputs separately; never join the branches through zero. More points are needed near the axes where the curve changes rapidly. Neither axis is reached for non-zero a.
To solve f(x) = c graphically, intersect y = f(x) with y = c and read x. To solve f(x) = g(x), graph both functions and read the intersection inputs. Graph readings are approximate unless exact values are separately justified.
Sources for this lesson
Pause and explain
What does the input at an intersection of y = f(x) and y = g(x) solve?
Worked example
Table y = x² − 2x − 3 at x = −1, 0, 1, 2, 3. Where does y = 0?
Show the worked solution
- The outputs are 0, −3, −4, −3, 0.
- Plot the points and draw a smooth upward parabola with its lowest point at (1, −4).
- The graph crosses the x-axis at −1 and 3; substitution confirms the readings exactly.
Answer Table: 0, −3, −4, −3, 0; roots x = −1 and x = 3.
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.
Table y = −x² + 4 at x = −2, −1, 0, 1, 2, then describe its graph.
Give me a hint
Negate the square after calculating it.
Compare my reasoning
- Outputs are 0, 3, 4, 3, 0.
- The parabola opens down, is symmetric about x = 0 and has maximum point (0, 4).
0, 3, 4, 3, 0; downward parabola through (−2, 0) and (2, 0), maximum (0, 4).
Look for these in your work
- Use negative inputs correctly
- Plot with labelled scales
Table y = 6/x for x = −3, −2, −1, 1, 2, 3. Explain the missing input 0.
Give me a hint
Divide 6 by each allowed input.
Compare my reasoning
- Outputs are −2, −3, −6, 6, 3, 2.
- Division by zero is undefined, so there are two separated branches.
−2, −3, −6, 6, 3, 2; omit x = 0 and do not connect across it.
Look for these in your work
- Retain the signs
- Show the domain break
Sketch y = x² and y = x + 2 for −2 ≤ x ≤ 3. Read their intersection inputs, then verify them.
Give me a hint
Compare the outputs in a table; solve x² = x + 2 as a check.
Compare my reasoning
- Both curves pass through (−1, 1) and (2, 4).
- x² − x − 2 = (x − 2)(x + 1) = 0 confirms the graph readings.
x ≈ −1 and x ≈ 2 from the graph; exact checks give x = −1 and 2.
Look for these in your work
- Read inputs, not outputs
- Distinguish graph estimates from exact verification
Common mistakes
- Do not connect a reciprocal graph through x = 0.
- Joining a quadratic table with straight segments is not the intended smooth curve.
What can you explain now?
What does the input at an intersection of y = f(x) and y = g(x) solve?
Compare with the explanation
f(x) = g(x).
The two outputs are equal at the same input.
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Make it stick
What does the input at an intersection of y = f(x) and y = g(x) solve?
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.