Transformations and Derivative Graphs
Track horizontal and vertical transformations and infer the sign and zeros of f′ from the behaviour of f.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
vertical scaling
Exam cue: Does the change act inside or outside f?
Concept 2
horizontal scaling
Exam cue: Which coordinates stay fixed under the transformation?
Concept 3
translations
Exam cue: Where is f increasing, decreasing or stationary?
Concept 4
reflections
Concept 5
qualitative derivative
Risk pitfalls and guardrails
Using the same direction rule for f(x + a) and f(x) + a
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Forgetting the reciprocal horizontal scale
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Sketching f′ with the same shape as f
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Outside is ordinary
An outside vertical change follows its visible sign and factor.
Inside is inverse
Horizontal shifts and scales act opposite to the visible input change.
Order affects combined moves
Transform coordinates one operation at a time.
Rising means positive
Where f increases, f′ lies above the axis.
Turning means zero
A differentiable stationary point gives f′ = 0.
Slope, not height
The derivative graph records gradient rather than the y-value of f.
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 point (2, 5) lies on y=f(x). Which point lies on y=f(x)+3?
The point (−1, 4) lies on y=f(x). Which point lies on y=f(x−5)?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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