Generalization, model comparison, proof, and technology-assisted validation.
Directions: Show exact work before decimals, label domains or intervals, and explain calculator use.
1. {'id': 'u02-l07', 'number': 7, 'title': 'Polynomial and Rational Modelling', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
State the domain of f(x)=(x+1)/(x−4).
2. {'id': 'u02-l07', 'number': 7, 'title': 'Polynomial and Rational Modelling', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Find the vertical asymptote of f(x)=3/(x+2).
3. {'id': 'u02-l07', 'number': 7, 'title': 'Polynomial and Rational Modelling', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Find the horizontal asymptote of (2x²+1)/(x²−9).
4. {'id': 'u02-l07', 'number': 7, 'title': 'Polynomial and Rational Modelling', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Simplify (x²−9)/(x−3) and identify the excluded value.
5. {'id': 'u02-l07', 'number': 7, 'title': 'Polynomial and Rational Modelling', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Solve 1/x+1/(x+2)=1.
6. {'id': 'u02-l07', 'number': 7, 'title': 'Polynomial and Rational Modelling', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
On which interval is 1/(x−5) positive?
7. {'id': 'u02-l08', 'number': 8, 'title': 'Optimization and Technology-Assisted Validation', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
State the domain of f(x)=(x+1)/(x−4).
8. {'id': 'u02-l08', 'number': 8, 'title': 'Optimization and Technology-Assisted Validation', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Find the vertical asymptote of f(x)=3/(x+2).
9. {'id': 'u02-l08', 'number': 8, 'title': 'Optimization and Technology-Assisted Validation', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Find the horizontal asymptote of (2x²+1)/(x²−9).
10. {'id': 'u02-l08', 'number': 8, 'title': 'Optimization and Technology-Assisted Validation', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Simplify (x²−9)/(x−3) and identify the excluded value.
11. {'id': 'u02-l08', 'number': 8, 'title': 'Optimization and Technology-Assisted Validation', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
Solve 1/x+1/(x+2)=1.
12. {'id': 'u02-l08', 'number': 8, 'title': 'Optimization and Technology-Assisted Validation', 'family': 'rational', 'week': 6, 'unit': 2, 'unit_title': 'Polynomial & Rational Functions', 'essential': 'How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?', 'project': 'Polynomial-Rational Design Specification'}
On which interval is 1/(x−5) positive?