MathematicsChapter 1: Functions and Graphs
Polynomial and Rational Graphs
Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
Learning objectives
- State the rule for polynomial and rational graphs.
- Solve a polynomial and rational graphs problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
Formula
x=-1 is a zero and x
Where / symbols
- Polynomial and rational graphs
- Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
CONCEPT AND RULE
Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
METHOD
For polynomial and rational graphs, 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
For x+1x-2, x=-1 is a zero and x=2 is a vertical asymptote.
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 polynomial and rational graphs
- 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:
- For x+1x-2, x=-1 is a zero and x=2 is a vertical asymptote.
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
- Polynomial and rational graphs
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Polynomial and Rational Graphs
Sticky Notes · Formula and Worked Revision
Core Idea
- Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
Rule / Formula
- Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For x+1x-2, x=-1 is a zero and x=2 is a vertical asymptote.
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
- Zeros give x-intercepts; denominator zeros mark exclusions and possible vertical asymptotes.
- Worked pattern: For x+1x-2, x=-1 is a zero and x=2 is a vertical asymptote.
- Final check: domain, sign, notation, units, and reasonableness.
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