GRADE 12 MATHEMATICS
QUICK & INFORMATIVE REVIEWER
Easy-to-understand summary • Key concepts • Formulas • Examples
1. FUNCTIONS & THEIR GRAPHS
Function: a rule that assigns exactly ONE output y to each input x. Written f(x).
Domain = allowed x-values. Range = resulting y-values.
Evaluate by substituting the given x into the function.
Composition: (f ∘ g)(x) = f(g(x)). Do g first, then f.
Inverse function reverses a function: f⁻¹(f(x)) = x. To find it, write y=f(x), swap x and y, then solve for y.
Example: f(x)=2x+3 → f(4)=2(4)+3=11
2. POLYNOMIAL & RATIONAL FUNCTIONS
Polynomial: expression with non-negative integer exponents, e.g., 3x³−2x+1.
Degree = highest exponent. Leading coefficient = coefficient of the highest-degree term.
Zeros/roots are x-values where f(x)=0; they are x-intercepts of the graph.
Rational function: f(x)=P(x)/Q(x). Domain excludes values that make Q(x)=0.
Vertical asymptote often occurs where the denominator is zero after simplifying; horizontal asymptote describes end
behavior.
Factoring: x²−a²=(x−a)(x+a)
Quadratic formula: x = [−b ± √(b²−4ac)]/(2a) for ax²+bx+c=0
3. EXPONENTIAL & LOGARITHMIC FUNCTIONS
Exponential: f(x)=abˣ, where a≠0, b>0, b≠1. Growth if b>1; decay if 0<b<1.
Logarithm answers: 'What exponent gives this number?' log_b(x)=y means bʸ=x.
Log laws: log_b(MN)=log_b M + log_b N; log_b(M/N)=log_b M−log_b N; log_b(Mᵏ)=k log_b M.
Change of base: log_b a = log(a)/log(b).
Exponential and logarithmic functions are inverses.
Example: log₂(8)=3 because 2³=8
4. SEQUENCES & SERIES
Arithmetic sequence: common difference d. aₙ=a₁+(n−1)d.
Arithmetic series: Sₙ=n/2 [2a₁+(n−1)d].
Geometric sequence: common ratio r. aₙ=a₁rⁿ⁻¹.
Finite geometric series: Sₙ=a₁(1−rⁿ)/(1−r), r≠1.
Infinite geometric series converges when |r|<1: S∞=a ₁/(1−r).
Example: 3, 7, 11, 15,... is arithmetic with d=4
5. FINANCIAL MATHEMATICS
Simple interest: interest is based only on the original principal.
Compound interest: interest is added to the balance, so later interest can earn interest.
Always match the interest rate period with the time period (e.g., monthly rate with months).
Simple interest: I=Prt; Amount A=P+I
Compound interest: A=P(1+r/n)ⁿᵗ
Present value: P=A/(1+r/n)ⁿᵗ
6. PROBABILITY
Probability measures how likely an event is: 0≤P(A)≤1.
Complement: P(Aᶜ)=1−P(A).
Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B).
If events are independent: P(A∩B)=P(A)P(B).
Conditional probability: P(A|B)=P(A∩B)/P(B), when P(B)>0.
Counting: n! = n(n−1)(n−2)...1.
Grade 12 Mathematics Reviewer
QUICK & INFORMATIVE REVIEWER
Easy-to-understand summary • Key concepts • Formulas • Examples
1. FUNCTIONS & THEIR GRAPHS
Function: a rule that assigns exactly ONE output y to each input x. Written f(x).
Domain = allowed x-values. Range = resulting y-values.
Evaluate by substituting the given x into the function.
Composition: (f ∘ g)(x) = f(g(x)). Do g first, then f.
Inverse function reverses a function: f⁻¹(f(x)) = x. To find it, write y=f(x), swap x and y, then solve for y.
Example: f(x)=2x+3 → f(4)=2(4)+3=11
2. POLYNOMIAL & RATIONAL FUNCTIONS
Polynomial: expression with non-negative integer exponents, e.g., 3x³−2x+1.
Degree = highest exponent. Leading coefficient = coefficient of the highest-degree term.
Zeros/roots are x-values where f(x)=0; they are x-intercepts of the graph.
Rational function: f(x)=P(x)/Q(x). Domain excludes values that make Q(x)=0.
Vertical asymptote often occurs where the denominator is zero after simplifying; horizontal asymptote describes end
behavior.
Factoring: x²−a²=(x−a)(x+a)
Quadratic formula: x = [−b ± √(b²−4ac)]/(2a) for ax²+bx+c=0
3. EXPONENTIAL & LOGARITHMIC FUNCTIONS
Exponential: f(x)=abˣ, where a≠0, b>0, b≠1. Growth if b>1; decay if 0<b<1.
Logarithm answers: 'What exponent gives this number?' log_b(x)=y means bʸ=x.
Log laws: log_b(MN)=log_b M + log_b N; log_b(M/N)=log_b M−log_b N; log_b(Mᵏ)=k log_b M.
Change of base: log_b a = log(a)/log(b).
Exponential and logarithmic functions are inverses.
Example: log₂(8)=3 because 2³=8
4. SEQUENCES & SERIES
Arithmetic sequence: common difference d. aₙ=a₁+(n−1)d.
Arithmetic series: Sₙ=n/2 [2a₁+(n−1)d].
Geometric sequence: common ratio r. aₙ=a₁rⁿ⁻¹.
Finite geometric series: Sₙ=a₁(1−rⁿ)/(1−r), r≠1.
Infinite geometric series converges when |r|<1: S∞=a ₁/(1−r).
Example: 3, 7, 11, 15,... is arithmetic with d=4
5. FINANCIAL MATHEMATICS
Simple interest: interest is based only on the original principal.
Compound interest: interest is added to the balance, so later interest can earn interest.
Always match the interest rate period with the time period (e.g., monthly rate with months).
Simple interest: I=Prt; Amount A=P+I
Compound interest: A=P(1+r/n)ⁿᵗ
Present value: P=A/(1+r/n)ⁿᵗ
6. PROBABILITY
Probability measures how likely an event is: 0≤P(A)≤1.
Complement: P(Aᶜ)=1−P(A).
Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B).
If events are independent: P(A∩B)=P(A)P(B).
Conditional probability: P(A|B)=P(A∩B)/P(B), when P(B)>0.
Counting: n! = n(n−1)(n−2)...1.
Grade 12 Mathematics Reviewer
