AUSTRALIA MATH YEAR 12 – EDmama Australia

A. Numbers and Counting

A.01. Evaluate numerical expressions using the correct order of operations.
A.02. Convert between decimals, fractions, and percentages fluently.
A.03. Understand and apply properties of integers (addition, subtraction, multiplication, division).
A.04. Perform operations with rational numbers (addition, subtraction, multiplication, division).
A.05. Simplify complex numerical expressions involving rational numbers, including mixed numbers and improper fractions.
A.06. Estimate and approximate values of irrational numbers in decimal form.
A.07. Explore the historical significance and real-world applications of rational and irrational numbers.

B. Functions

B.01. Evaluate functions for given inputs using function notation.
B.02. Find values of functions from their graphs and tables.
B.03. Complete a table for a function graph.
B.04. Identify functions from various representations (graphs, equations, tables).
B.05. Determine the domain and range of a function from its graph and equation.
B.06. Find the equation of a linear function from its graph or given points.
B.07. Graph linear functions, identifying slope and intercepts.
B.08. Find the gradient of a linear function.
B.09. Complete a function table for absolute value functions.
B.10. Graph absolute value functions.
B.11. Analyze linear functions over unit intervals.
B.12. Add, subtract, multiply, and divide functions.
B.13. Compose functions and determine the domain of the composite function.
B.14. Identify inverse functions algebraically and graphically.
B.15. Determine values of inverse functions from tables and graphs.
B.16. Find inverse functions and relations, stating any restrictions on the domain.
B.17. Sketch the graph of the inverse of a function.

C. Families of Functions

C.01. Identify and describe translations of functions.
C.02. Identify and describe reflections of functions across the x-axis and y-axis.
C.03. Identify and describe dilations (stretches and compressions) of functions.
C.04. Apply combinations of transformations to functions.
C.05. Use function transformation rules to predict the effects of transformations on a function’s graph and equation.
C.06. Given a graph, describe the transformations applied to a parent function.
C.07. Determine the equation of a transformed function given its graph.
C.08. Explore the symmetry properties (even and odd) of transformed functions.

D. Quadratic Functions

D.01. Identify key characteristics of quadratic functions (vertex, axis of symmetry, intercepts).
D.02. Graph quadratic functions in vertex form and standard form.
D.03. Match quadratic functions with their corresponding graphs.
D.04. Find the maximum or minimum value of a quadratic function.
D.05. Solve quadratic equations using square roots.
D.06. Solve quadratic equations by factoring.
D.07. Solve quadratic equations by completing the square.
D.08. Solve quadratic equations using the quadratic formula.
D.09. Use the discriminant to determine the nature and number of solutions to a quadratic equation.
D.10. Apply quadratic functions to model real-world scenarios, such as projectile motion and optimization problems.

E. Polynomials

E.01. Evaluate polynomials using synthetic division.
E.02. Find the roots of factored polynomials.
E.03. Divide polynomials using long division.
E.04. Divide polynomials using synthetic division.
E.05. Factorize sums and differences of cubes.
E.06. Solve equations with sums and differences of cubes.
E.07. Factorize using a quadratic pattern.
E.08. Solve equations using a quadratic pattern.
E.09. Write a polynomial from its roots (real and complex).
E.10. Apply the Rational Root Theorem to find potential rational roots.
E.11. Apply the Complex Conjugate Theorem to determine conjugate pairs of complex roots.
E.12. Utilize Descartes’ Rule of Signs to predict the number of positive and negative real roots.
E.13. State and apply the Fundamental Theorem of Algebra.
E.14. Match polynomials with their graphs.
E.15. Utilize Pascal’s triangle to determine binomial coefficients.
E.16. Expand binomials using Pascal’s triangle and the Binomial Theorem.
E.17. Apply the Binomial Theorem to find specific terms in a binomial expansion.
E.18. Apply polynomial functions to model real-world scenarios, such as curve fitting and optimization problems.

F. Roots and Rational Exponents

F.10. Evaluate expressions containing radicals and rational exponents.
F.01. Find roots of integers.
F.02. Find roots of rational numbers.
F.03. Find roots using a calculator.
F.04. Evaluate rational exponents.
F.05. Perform operations with rational exponents.
F.06. Understand and apply the concept of nth roots.
F.07. Simplify radical expressions with variables.
F.08. Simplify expressions involving rational exponents.
F.09. Convert between radical notation and exponential notation.

G. Exponential and Logarithmic Functions

G.01. Convert between exponential and logarithmic forms.
G.02. Evaluate logarithms using the definition of a logarithm.
G.03. State the domain and range of exponential and logarithmic functions.
G.04. Apply the change of base formula to evaluate logarithms with different bases.
G.05. Apply the product property of logarithms to simplify logarithmic expressions.
G.06. Apply the quotient property of logarithms to simplify logarithmic expressions.
G.07. Apply the power property of logarithms to simplify logarithmic expressions.
G.08. Evaluate logarithms using properties of logarithms.
G.09. Solve exponential equations by rewriting the base.
G.10. Solve exponential equations using logarithms.
G.11. Solve logarithmic equations with one logarithm.
G.12. Solve logarithmic equations with multiple logarithms.
G.13. Recognize and identify linear and exponential functions from tables, graphs, and equations.
G.14. Analyze exponential functions over unit intervals.
G.15. Describe linear and exponential growth and decay.
G.16. Solve exponential growth and decay word problems.
G.17. Solve compound interest word problems.
G.18. Apply exponential and logarithmic functions to model real-world scenarios, such as population growth, radioactive decay, and financial investments.

H. Radical Functions

I. Rational Functions

J. Simultaneous Equations

J.01. Solve simultaneous equations by graphing.
J.02. Solve simultaneous equations by graphing: word problems.
J.03. Classify simultaneous equations (independent, dependent, inconsistent).
J.04. Solve simultaneous equations using substitution.
J.05. Solve simultaneous equations using substitution: word problems.
J.06. Solve simultaneous equations using elimination.
J.07. Solve simultaneous equations using elimination: word problems.
J.08. Solve simultaneous equations in three variables using substitution.
J.09. Solve simultaneous equations in three variables using elimination.
J.10. Determine the number of solutions to simultaneous equations in three variables.
J.11. Apply simultaneous equations to model real-world scenarios, such as mixture problems or break-even analysis.

K. Inequalities and Linear Programming

K.01. Graph solutions to linear inequalities on a number line and in the coordinate plane.
K.02. Solve systems of linear inequalities by graphing.
K.03. Find the vertices of a solution set for a system of linear inequalities.
K.04. Apply linear programming to optimize a function subject to constraints.
K.05. Graph solutions to quadratic inequalities on a number line.
K.06. Solve quadratic inequalities algebraically.
K.07. Graph solutions to higher-degree inequalities on a number line.
K.08. Solve higher-degree inequalities algebraically.
K.09. Use inequalities to model real-world constraints and optimize solutions.

L. Matrices

L.01. Define matrix vocabulary (dimensions, elements, rows, columns).
L.02. State matrix operation rules (addition, subtraction, multiplication).
L.03. Add and subtract matrices.
L.04. Multiply a matrix by a scalar.
L.05. Perform linear combinations of matrices.
L.06. Multiply two matrices, verifying compatibility.
L.07. Simplify matrix expressions using the order of operations.
L.08. Solve matrix equations.
L.09. Calculate the determinant of a matrix (2×2 and 3×3).
L.10. Determine if a matrix is invertible (singular or non-singular).
L.11. Find the inverse of a matrix.
L.12. Verify if two matrices are inverses of each other.
L.13. Solve matrix equations using inverses.
L.14. Identify transformation matrices (rotation, reflection, dilation).
L.15. Write the vertex matrix for a geometric figure.
L.16. Apply transformation matrices to graph the image of a geometric figure.
L.17. Use matrices to represent and solve systems of linear equations.

M. Two-Dimensional Vectors

M.01. Find the magnitude of a vector.
M.02. Find the direction angle of a vector.
M.03. Find the component form of a vector.
M.04. Find the component form of a vector from its magnitude and direction angle.
M.05. Find a unit vector in the direction of a given vector.
M.06. Add and subtract vectors geometrically and algebraically.
M.07. Multiply a vector by a scalar.
M.08. Find the magnitude or direction of a vector scalar multiple.
M.09. Find the magnitude and direction of a vector sum (resultant vector).
M.10. Perform linear combinations of vectors.
M.11. Graph a resultant vector using the triangle method.
M.12. Graph a resultant vector using the parallelogram method.
M.13. Calculate the dot product of two vectors and interpret its geometric meaning.
M.14. Apply vectors to model real-world scenarios, such as forces, velocities, and displacements.

N. Three-Dimensional Vectors

N.01. Find the magnitude of a three-dimensional vector.
N.02. Find the component form of a three-dimensional vector.
N.03. Find a three-dimensional unit vector.
N.04. Add and subtract three-dimensional vectors.
N.05. Multiply three-dimensional vectors by a scalar.
N.06. Perform linear combinations of three-dimensional vectors.
N.07. Calculate the dot product of two three-dimensional vectors and interpret its geometric meaning.
N.08. Calculate the cross product of two three-dimensional vectors and interpret its geometric meaning.
N.09. Apply three-dimensional vectors to model real-world scenarios, such as forces, velocities, and displacements in three-dimensional space.

O. Trigonometry

O.01. Convert between radians and degrees.
O.02. Calculate arc length using radians.
O.03. Identify quadrants in the unit circle.
O.04. Find coterminal and reference angles.
O.05. Find trigonometric ratios (sine, cosine, tangent) using right triangles (SOH CAH TOA).
O.06. Find trigonometric ratios using the unit circle.
O.07. Find trigonometric ratios using reference angles.
O.08. Evaluate inverses of trigonometric functions (arcsin, arccos, arctan).
O.09. Solve trigonometric equations.
O.10. Use trigonometric ratios to find a side length in a right triangle.
O.11. Use trigonometric ratios to find an angle measure in a right triangle.
O.12. Solve a right triangle (find all sides and angles).
O.13. Apply the Law of Sines to solve triangles.
O.14. Apply the Law of Cosines to solve triangles.
O.15. Solve a triangle (find all sides and angles) given sufficient information.
O.16. Calculate the area of a triangle using the sine formula.
O.17. Calculate the area of a triangle using Heron’s formula.
O.18. Apply trigonometric concepts to model real-world scenarios, such as navigation, surveying, and oscillations.

P. Trigonometric Functions

P.01. Find properties of sine functions (amplitude, period, phase shift, vertical shift).
P.02. Write equations of sine functions from graphs.
P.03. Write equations of sine functions using properties.
P.04. Graph sine functions.
P.05. Find properties of cosine functions (amplitude, period, phase shift, vertical shift).
P.06. Write equations of cosine functions from graphs.
P.07. Write equations of cosine functions using properties.
P.08. Graph cosine functions.
P.09. Graph sine and cosine functions with various transformations.
P.10. Extend the concepts of sine and cosine functions to model periodic phenomena, such as sound waves, light waves, and oscillations.

Q. Trigonometric Identities

Q.05. Apply trigonometric sum and difference identities to simplify expressions and solve equations.
Q.06. Apply double angle and half angle identities.
Q.07. Verify trigonometric identities using algebraic manipulation and other known identities.
Q.08. Simplify complex trigonometric expressions using identities.
Q.01. State and apply complementary angle identities.
Q.02. State and apply identities related to the symmetry and periodicity of trigonometric functions.
Q.03. Find trigonometric ratios using Pythagorean or reciprocal identities.
Q.04. Find trigonometric ratios using multiple identities.

R. Conic Sections

R.01. Find properties of parabolas (vertex, focus, directrix, axis of symmetry).
R.02. Write equations of parabolas in vertex form.
R.03. Graph parabolas.
R.04. Find properties of circles (center, radius).
R.05. Write equations of circles in standard form.
R.06. Graph circles.
R.07. Find properties of ellipses (center, foci, vertices, major axis, minor axis).
R.08. Find the eccentricity of an ellipse.
R.09. Write equations of ellipses in standard form.
R.10. Find properties of hyperbolas (center, foci, vertices, transverse axis, conjugate axis, asymptotes).
R.11. Find the eccentricity of a hyperbola.
R.12. Write equations of hyperbolas in standard form.
R.13. Convert equations of conic sections from general to standard form.
R.14. Identify conic sections from their equations in general form.
R.15. Apply conic sections to model real-world scenarios, such as planetary orbits, satellite dishes, and architectural designs.

S. Complex Numbers

S.01. Define the imaginary unit ‘i’ and understand its properties.
S.02. Add and subtract complex numbers.
S.03. Find complex conjugates.
S.04. Multiply and divide complex numbers.
S.05. Perform addition, subtraction, multiplication, and division with complex numbers.
S.06. Find the absolute values of complex numbers.
S.07. Simplify powers of i.
S.08. Convert complex numbers between rectangular and polar forms.
S.09. Apply complex numbers to solve quadratic equations with no real solutions.

T. Complex Plane

T.01. Define the complex plane and its axes (real and imaginary).
T.02. Graph complex numbers in the complex plane.
T.03. Perform addition and subtraction of complex numbers graphically in the complex plane.
T.04. Graph complex conjugates in the complex plane and observe their symmetry.
T.05. Calculate the absolute value of a complex number geometrically in the complex plane.
T.06. Find midpoints in the complex plane.
T.07. Calculate distance in the complex plane.
T.08. Use the complex plane to visualize and analyze complex number operations.

U. Polar Form

U.01. Find the modulus (magnitude) and argument (angle) of a complex number.
U.02. Convert complex numbers from rectangular to polar form.
U.03. Convert complex numbers from polar to rectangular form.
U.04. Convert complex numbers between rectangular and polar form.
U.05. Match polar equations and graphs.
U.06. Perform multiplication and division of complex numbers in polar form.
U.07. Apply DeMoivre’s Theorem to find powers and roots of complex numbers in polar form.
U.08. Represent complex numbers in exponential form (Euler’s formula).

V. Sequences and Series

V.01. Find terms of a sequence given an explicit formula.
V.02. Find terms of a sequence given a recursive formula.
V.03. Classify a sequence as explicit or recursive.
V.04. Find a recursive formula for a given sequence.
V.05. Find both recursive and explicit formulas for a given sequence.
V.06. Convert a recursive formula to an explicit formula.
V.07. Convert an explicit formula to a recursive formula.
V.08. Convert between explicit and recursive formulas.
V.09. Understand and use sigma notation to represent series.
V.10. Identify arithmetic and geometric series.
V.11. Find the sum of a finite arithmetic or geometric series.
V.12. Define and understand the concept of partial sums.
V.13. Calculate partial sums of arithmetic series.
V.14. Calculate partial sums of geometric series.
V.15. Review partial sums of both arithmetic and geometric series.
V.16. Determine if a geometric series is convergent or divergent.
V.17. Find the value of an infinite geometric series.
V.18. Write a repeating decimal as a fraction using the formula for an infinite geometric series.
V.19. Apply sequences and series to model real-world scenarios, such as compound interest, annuities, and amortization.

W. Probability

W.01. Define probability and related terminology (event, sample space, outcome).
W.02. Calculate probabilities of events.
W.03. Calculate combinations and permutations.
W.04. Find probabilities using combinations and permutations.
W.05. Find probabilities using two-way frequency tables.
W.06. Define and identify independent events.
W.07. Calculate conditional probabilities.
W.08. Determine if events are independent using conditional probability.
W.09. Find conditional probabilities using two-way frequency tables.
W.10. Find probabilities using the addition rule for mutually exclusive and non-mutually exclusive events.
W.11. Calculate probabilities of compound events using various probability rules.

X. Probability Distributions

X.01. Define and identify discrete and continuous random variables.
X.02. Write a discrete probability distribution.
X.03. Graph a discrete probability distribution.
X.04. Calculate expected values of random variables.
X.05. Calculate the variance of random variables.
X.06. Calculate the standard deviation of random variables.
X.07. Write the probability distribution for a game of chance.
X.08. Calculate expected values for a game of chance.
X.09. Determine which bet is better based on expected value.
X.10. Find probabilities using the binomial distribution.
X.11. Calculate the mean, variance, and standard deviation of binomial distributions.
X.12. Find probabilities using the normal distribution I (using z-scores).
X.13. Find probabilities using the normal distribution II (using z-tables or calculators).
X.14. Find z-values (z-scores) corresponding to given probabilities.
X.15. Find values of normal variables corresponding to given probabilities.
X.16. Understand distributions of sample means and the concept of standard error.
X.17. State and apply the Central Limit Theorem.
X.18. Use normal distributions to approximate binomial distributions.
X.19. Apply probability distributions to model real-world scenarios, such as quality control, insurance, and stock market analysis.

Y. Statistics

Y.01. Define population, sample, parameter, and statistic.
Y.02. Identify biased samples and sources of bias in data collection.
Y.03. Calculate variance and standard deviation for a data set.
Y.04. Identify an outlier in a data set.
Y.05. Identify an outlier and describe the effect of removing it on the mean and standard deviation.
Y.06. Identify outliers in scatter plots.
Y.07. Match correlation coefficients to scatter plots.
Y.08. Calculate correlation coefficients (Pearson’s r).
Y.09. Find the equation of a regression line (least squares regression line).
Y.10. Interpret the slope and y-intercept of a regression line.
Y.11. Analyze a regression line of a data set, including assessing its goodness of fit.
Y.12. Analyze a regression line using statistics of a data set (mean, standard deviation, correlation coefficient).
Y.13. Find confidence intervals for population means using t-distributions or z-distributions.
Y.14. Find confidence intervals for population proportions.
Y.15. Interpret confidence intervals for population means and proportions.
Y.16. Conduct hypothesis tests for population means and proportions.

Z. Introduction to Derivatives

Z.01. Calculate the average rate of change of a function over an interval I.
Z.02. Calculate the average rate of change of a function over an interval II.
Z.03. Find instantaneous rates of change using limits.
Z.04. Interpret velocity as a rate of change.
Z.05. Find values of derivatives using limits.
Z.06. Find the gradient of a tangent line using limits.
Z.07. Find equations of tangent lines using limits.
Z.08. Define and interpret the derivative as the slope of the tangent line and the instantaneous rate of change.

AA. Derivative Rules

BB. Calculate Derivatives

BB.01. Find derivatives of polynomials.
BB.02. Find derivatives of rational functions.
BB.03. Find derivatives of trigonometric functions I (sine, cosine).
BB.04. Find derivatives of trigonometric functions II (tangent, cotangent, secant, cosecant).
BB.05. Find derivatives of exponential functions.
BB.06. Find derivatives of logarithmic functions.
BB.07. Find derivatives of inverse trigonometric functions.
BB.08. Find derivatives of radical functions.
BB.09. Find derivatives using the product rule I (simpler functions).
BB.10. Find derivatives using the product rule II (more complex functions).
BB.11. Find derivatives using the quotient rule I (simpler functions).
BB.12. Find derivatives using the quotient rule II (more complex functions).
BB.13. Find derivatives using the chain rule I (simpler composite functions).
BB.14. Find derivatives using the chain rule II (more complex composite functions).

Course Content

A. Numbers and Counting

A.01. Evaluate numerical expressions using the correct order of operations.

A.02. Convert between decimals, fractions, and percentages fluently.

A.03. Understand and apply properties of integers (addition, subtraction, multiplication, division).

A.04. Perform operations with rational numbers (addition, subtraction, multiplication, division).

A.05. Simplify complex numerical expressions involving rational numbers, including mixed numbers and improper fractions.

A.06. Estimate and approximate values of irrational numbers in decimal form.

A.07. Explore the historical significance and real-world applications of rational and irrational numbers.

B. Functions

B.01. Evaluate functions for given inputs using function notation.

B.02. Find values of functions from their graphs and tables.

B.03. Complete a table for a function graph.

B.04. Identify functions from various representations (graphs, equations, tables).

B.05. Determine the domain and range of a function from its graph and equation.

B.06. Find the equation of a linear function from its graph or given points.

B.07. Graph linear functions, identifying slope and intercepts.

B.08. Find the gradient of a linear function.

B.09. Complete a function table for absolute value functions.

B.10. Graph absolute value functions.

B.11. Analyze linear functions over unit intervals.

B.12. Add, subtract, multiply, and divide functions.

B.13. Compose functions and determine the domain of the composite function.

B.14. Identify inverse functions algebraically and graphically.

B.15. Determine values of inverse functions from tables and graphs.

B.16. Find inverse functions and relations, stating any restrictions on the domain.

B.17. Sketch the graph of the inverse of a function.

C. Families of Functions

C.01. Identify and describe translations of functions.

C.02. Identify and describe reflections of functions across the x-axis and y-axis.

C.03. Identify and describe dilations (stretches and compressions) of functions.

C.04. Apply combinations of transformations to functions.

C.05. Use function transformation rules to predict the effects of transformations on a function’s graph and equation.

C.06. Given a graph, describe the transformations applied to a parent function.

C.07. Determine the equation of a transformed function given its graph.

C.08. Explore the symmetry properties (even and odd) of transformed functions.

D. Quadratic Functions

D.01. Identify key characteristics of quadratic functions (vertex, axis of symmetry, intercepts).

D.02. Graph quadratic functions in vertex form and standard form.

D.03. Match quadratic functions with their corresponding graphs.

D.04. Find the maximum or minimum value of a quadratic function.

D.05. Solve quadratic equations using square roots.

D.06. Solve quadratic equations by factoring.

D.07. Solve quadratic equations by completing the square.

D.08. Solve quadratic equations using the quadratic formula.

D.09. Use the discriminant to determine the nature and number of solutions to a quadratic equation.

D.10. Apply quadratic functions to model real-world scenarios, such as projectile motion and optimization problems.

E. Polynomials

E.01. Evaluate polynomials using synthetic division.

E.02. Find the roots of factored polynomials.

E.03. Divide polynomials using long division.

E.04. Divide polynomials using synthetic division.

E.05. Factorize sums and differences of cubes.

E.06. Solve equations with sums and differences of cubes.

E.07. Factorize using a quadratic pattern.

E.08. Solve equations using a quadratic pattern.

E.09. Write a polynomial from its roots (real and complex).

E.10. Apply the Rational Root Theorem to find potential rational roots.

E.11. Apply the Complex Conjugate Theorem to determine conjugate pairs of complex roots.

E.12. Utilize Descartes’ Rule of Signs to predict the number of positive and negative real roots.

E.13. State and apply the Fundamental Theorem of Algebra.

E.14. Match polynomials with their graphs.

E.15. Utilize Pascal’s triangle to determine binomial coefficients.

E.16. Expand binomials using Pascal’s triangle and the Binomial Theorem.

E.17. Apply the Binomial Theorem to find specific terms in a binomial expansion.

E.18. Apply polynomial functions to model real-world scenarios, such as curve fitting and optimization problems.

F. Roots and Rational Exponents

F.10. Evaluate expressions containing radicals and rational exponents.

F.01. Find roots of integers.

F.02. Find roots of rational numbers.

F.03. Find roots using a calculator.

F.04. Evaluate rational exponents.

F.05. Perform operations with rational exponents.

F.06. Understand and apply the concept of nth roots.

F.07. Simplify radical expressions with variables.

F.08. Simplify expressions involving rational exponents.

F.09. Convert between radical notation and exponential notation.

G. Exponential and Logarithmic Functions

G.01. Convert between exponential and logarithmic forms.

G.02. Evaluate logarithms using the definition of a logarithm.