Graphs of Functions
Sketch common functions, apply transformations and combine algebra and calculus to infer intersections and shape.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Recognise lines, polynomials, trigonometric, logarithmic, exponential, root and modulus graphs.
Exam cue: Start from domain, range, intercepts and asymptotic behaviour.
Concept 2
Apply vertical and horizontal translations, stretches, reflections and compositions.
Exam cue: Separate changes to the input from changes to the output.
Concept 3
Use algebra and differentiation to locate intercepts, roots, intersections and monotonic intervals.
Exam cue: Cross-check exact algebraic results against the proposed sketch.
Risk pitfalls and guardrails
Moving f(x+a) in the wrong direction.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Ignoring an excluded domain or asymptote.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Assuming a polynomial degree equals its number of real roots.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Outside change
af(x) and f(x)+a transform vertical outputs.
Inside change
f(x+a) and f(ax) transform horizontal inputs inversely.
Modulus output
|f(x)| reflects negative output above the axis.
Composition order
f(g(x)) applies g first and then f.
Intercepts
Set y = 0 for x-intercepts and x = 0 for the y-intercept.
Graph synthesis
Use algebra for exact points and calculus for local direction.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
The graph y=f(x) has roots −2 and 5. What is the distance between the roots of y=f(x−3)?
If f(x)=|x−3|+2, what is the minimum of f(x)+x for x≥3?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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