MathematicsChapter 1: Functions and Graphs
Basic Graph Transformations
In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
Learning objectives
- State the rule for basic graph transformations.
- Solve a basic graph transformations problem step by step.
- Check the result, restrictions, notation, and units.
Definition
In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
Mathematical form
y=(x-3)2+2 is y=x2 shifted 3 right and 2 up.
y=a f(b(x-h))+k
y=(x-3)2+2 is y
Where / symbols
- Basic graph transformations
- In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
CONCEPT AND RULE
In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
METHOD
For basic graph transformations, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
y=(x-3)2+2 is y=x2 shifted 3 right and 2 up.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve basic graph transformations
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- y=(x-3)2+2 is y=x2 shifted 3 right and 2 up.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Basic graph transformations
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Basic Graph Transformations
Sticky Notes · Formula and Worked Revision
Core Idea
- In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
Rule / Formula
- In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- y=(x-3)2+2 is y=x2 shifted 3 right and 2 up.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- In y=a f(b(x-h))+k, h and k shift the graph; a and b control stretch and reflection.
- Worked pattern: y=(x-3)^2+2 is y=x^2 shifted 3 right and 2 up.
- Final check: domain, sign, notation, units, and reasonableness.
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