Graph cubic, reciprocal-square and exponential functions
Graph rational multiples of powers and sums of up to three terms · Recognise cubic, reciprocal-square and exponential features · Use intersections for approximate solutions
Section E3.5 · Extended · Graph features, symmetry and end behaviour
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
- Core function graphs
- Negative integer powers and fractions
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
- Graph rational multiples of powers and sums of up to three terms
- Recognise cubic, reciprocal-square and exponential features
- Use intersections for approximate solutions
Extend the graph toolkit
Extended power graphs use rational coefficients and exponents −2, −1, 0, 1, 2 and 3; sums of up to three such terms can be graphed by tables. x⁻¹ = 1/x and x⁻² = 1/x² exclude zero. Plot separately on either side of a forbidden input.
The graph y = x³ is odd-symmetric about the origin, increases on all real inputs and has no maximum or minimum. y = 2/x² is even-symmetric, always positive, rises on its negative branch and falls on its positive branch; it approaches zero at the far ends.
Exponential outputs
For y = aˣ with positive integer base a > 1, the domain is all real inputs and the range is positive outputs. The graph passes through (0, 1), increases, and approaches zero as x becomes very negative. The base a = 1 is the constant function 1.
Make a table around useful inputs and refine near an intersection. A finite table is a plotting aid, not proof of all features. Use the formula's sign, symmetry or growth rule to justify what lies beyond the window.
Sources for this lesson
Pause and explain
What is the domain of y = 2/x²?
Worked example
Compare y = x³ and y = 2ˣ at x = −1, 0, 1, 2, 3.
Show the worked solution
- Cubic outputs: −1, 0, 1, 8, 27.
- Exponential outputs: 1/2, 1, 2, 4, 8.
- A crossing lies between 1 and 2 since the order reverses; more values refine the approximate crossing.
Answer The table gives one crossing between x = 1 and 2, not an exact root or a claim that there are no other crossings.
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 = 2/x² at x = −2, −1, 1, 2. State its symmetry and range on all allowed real inputs.
Give me a hint
Square the denominator before dividing.
Compare my reasoning
- Outputs are 1/2, 2, 2, 1/2.
- Equal outputs at opposite inputs give y-axis symmetry. All values are positive; any positive output is obtained at a suitable non-zero x.
1/2, 2, 2, 1/2; y-axis symmetry; range y > 0.
Look for these in your work
- Keep both branches
- Exclude zero from domain and range
Table y = (1/2)x³ − x + 2 at x = −2, −1, 0, 1, 2. Is it odd-symmetric about the origin?
Give me a hint
The constant term affects the symmetry test.
Compare my reasoning
- Outputs are 0, 5/2, 2, 3/2, 4.
- At x = 0, y = 2, not 0; also f(1) ≠ −f(−1).
0, 5/2, 2, 3/2, 4; it is not odd-symmetric about the origin.
Look for these in your work
- Combine all three terms
- Test rather than assume symmetry
For y = 3ˣ, find outputs at −1, 0 and 2; use these and the growth rule to describe intercepts and end behaviour.
Give me a hint
3⁻¹ is a reciprocal.
Compare my reasoning
- Outputs are 1/3, 1 and 9.
- There is a y-intercept (0, 1), no x-intercept and positive outputs. Multiplication by 3 per unit input gives unbounded growth as x increases; backwards it approaches 0.
1/3, 1, 9; y-intercept (0, 1), no x-intercept; grows to the right and approaches zero to the left.
Look for these in your work
- Interpret negative powers
- Explain beyond the table
Common mistakes
- x⁻² is positive for negative x; it is not −x².
- A plotting window cannot certify that all intersections have been found.
What can you explain now?
What is the domain of y = 2/x²?
Compare with the explanation
All real x except zero.
The squared denominator still vanishes at zero.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What is the domain of y = 2/x²?
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.