Function Graphs and Synthesis
Recognise and transform common graphs and combine algebra, calculus and intersections to determine curve behaviour.
How to prepare for the current ESAT
Confirm your course combination, build no-calculator fluency from Mathematics 1 upward, and use official Pearson materials for authentic timing and interface rehearsal.
Core concepts
Concept 1
Sketch lines, polynomials, trigonometric, logarithmic, exponential, square-root and modulus functions.
Exam cue: Start from anchor points, asymptotes, domain and range.
Concept 2
Apply horizontal and vertical translations, stretches, reflections and compositions.
Exam cue: Apply transformations in the correct input-versus-output direction.
Concept 3
Use algebra and differentiation to locate intercepts, intersections, stationary points and increasing or decreasing intervals.
Exam cue: Cross-check algebraic roots against the graph.
Risk pitfalls and guardrails
Moving a graph in the wrong direction for f(x + a).
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Ignoring domain restrictions or asymptotes.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Assuming every polynomial has as many real roots as its degree.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
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
|f(x)| reflects negative output above the axis.
Composition
f(g(x)) applies g first, then f.
Intercept
Set y = 0 for x-intercepts and x = 0 for the y-intercept.
Synthesis
Use algebra for exact points and calculus for local shape.
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
What vertical movement maps y=f(x) onto y=f(x)+4?
Which horizontal shift produces the graph y=f(x−4) from y=f(x)?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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