Newfoundland and Labrador

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Skills available for Newfoundland and Labrador grade 12 math curriculum

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1 Polynomial Functions

  • 1.RF Relations and Functions

    • 1.RF10 Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

      • 1.RF10.1 Explain how long division of a polynomial expression by a binomial expression of the form x - a, a ∈ I is related to synthetic division.

      • 1.RF10.2 Divide a polynomial expression by a binomial expression of the form x - a, a ∈ I using long division or synthetic division.

      • 1.RF10.3 Explain the relationship between the remainder when a polynomial expression is divided by x - a, a ∈ I, and the value of the polynomial expression at x = a (remainder theorem).

      • 1.RF10.4 Explain and apply the factor theorem to express a polynomial expression as a product of factors.

      • 1.RF10.5 Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function.

    • 1.RF11 Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5).

2 Function Transformations

  • 2.RF Relations and Functions

    • 2.RF1 Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations.

      • 2.RF1.1 Compare the graphs of a set of functions of the form y - k = f(x) to the graph of y = f(x) and generalize, using inductive reasoning, a rule about the effect of k.

      • 2.RF1.2 Compare the graphs of a set of functions of the form y = f(x - h) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effect of h.

      • 2.RF1.3 Compare the graphs of a set of functions of the form y - k = f(x - h) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effect of h and k.

      • 2.RF1.4 Sketch the graph of y - k = f(x), y = f (x - h) or y - k = f(x - h) for given values of h and k, given a sketch of the function y = f(x), where the equation of y = f(x) is not given.

      • 2.RF1.5 Write the equation of a function whose graph is a vertical and/or horizontal translation of the graph of the function y = f(x).

    • 2.RF2 Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations.

      • 2.RF2.1 Compare the graphs of a set of functions of the form y = af(x) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effect of a.

      • 2.RF2.2 Compare the graphs of a set of functions of the form y = f(bx) to the graph of y = f(x) and generalize, using inductive reasoning, a rule about the effect of b.

      • 2.RF2.3 Compare the graphs of a set of functions of the form y = af(bx) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effects of a and b.

      • 2.RF2.4 Sketch the graph of y = af(x), y = f(bx) or y = af(bx) for given values of a and b, given a sketch of the function y = f(x), where the equation of y = f(x) is not given.

      • 2.RF2.5 Write the equation of a function, given its graph which is a vertical and/or horizontal stretch of the graph of the function y = f(x).

    • 2.RF3 Apply translations and stretches to the graphs and equations of functions.

      • 2.RF3.1 Sketch the graph of the function y - k = af(b(x - h)) for given values of a, b, h and k, given the graph of the function y = f(x), where the equation of y = f(x) is not given.

      • 2.RF3.2 Write the equation of a function, given its graph which is a translation and/or stretch of the graph of the function y = f(x).

    • 2.RF4 Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the:

      • 2.RF4.a x-axis

      • 2.RF4.b y-axis

      • 2.RF4.c line y = x.

      • 2.RF4.1 Generalize the relationship between the coordinates of an ordered pair and the coordinates of the corresponding ordered pair that results from a reflection through the x-axis or the y-axis.

      • 2.RF4.2 Sketch the reflection of the graph of a function y = f(x) through the x-axis or the y-axis, given the graph of the function y = f(x), where the equation of y = f(x) is not given.

      • 2.RF4.3 Generalize, using inductive reasoning, and explain rules for the reflection of the graph of the function y = f(x) through the x-axis or the y-axis.

      • 2.RF4.4 Sketch the graphs of the functions y = -f(x) and y = f(-x), given the graph of the function y = f(x), where the equation of y = f(x) is not given.

      • 2.RF4.5 Write the equation of a function, given its graph which is a reflection of the graph of the function y = f(x) through the x-axis or the y-axis.

      • 2.RF4.6 Generalize the relationship between the coordinates of an ordered pair and the coordinates of the corresponding ordered pair that results from a reflection through the line y = x.

      • 2.RF4.7 Sketch the reflection of the graph of a function y = f(x) through the line y = x, given the graph of the function y = f(x), where the equation of y = f(x) is not given.

      • 2.RF4.8 Generalize, using inductive reasoning, and explain rules for the reflection of the graph of the function y = f(x) through the line y = x.

    • 2.RF5 Demonstrate an understanding of inverses of relations.

      • 2.RF5.1 Explain how the transformation (x, y) → (y, x) can be used to sketch the inverse of a relation.

      • 2.RF5.2 Explain the relationship between the domains and ranges of a relation and its inverse.

      • 2.RF5.3 Explain how the graph of the line y = x can be used to sketch the inverse of a relation.

      • 2.RF5.4 Sketch the graph of the inverse relation, given the graph of a relation.

      • 2.RF5.5 Determine if a relation and its inverse are functions.

      • 2.RF5.6 Determine restrictions on the domain of a function in order for its inverse to be a function.

      • 2.RF5.7 Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation.

      • 2.RF5.8 Determine, algebraically or graphically, if two functions are inverses of each other.

3 Radical Functions

  • 3.RF Relations and Functions

    • 3.RF12 Graph and analyze radical functions (limited to functions involving one radical).

      • 3.RF12.1 Sketch the graph of the function y = √x, using a table of values, and state the domain and range.

      • 3.RF12.2 Sketch the graph of the function y - k = a√ [b(x - h)] by applying transformations to the graph of the function y = √x, and state the domain and range.

      • 3.RF12.3 Sketch the graph of the function y = √f(x), given the equation or graph of the function y = f(x), and explain the strategies used.

      • 3.RF12.4 Compare the domain and range of the function y =√f(x), to the domain and range of the function y = f(x), and explain why the domains and ranges may differ.

      • 3.RF12.5 Describe the relationship between the roots of a radical equation and the x-intercepts of the graph of the corresponding radical function.

      • 3.RF12.6 Determine, graphically, an approximate solution of a radical equation.

4 Trigonometry and the Unit Circle

  • 4.T Trigonometry

    • 4.T1 Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

      • 4.T1.1 Sketch, in standard position, an angle (positive or negative) when the measure is given in degrees.

      • 4.T1.2 Sketch, in standard position, an angle with a measure of 1 radian.

      • 4.T1.3 Describe the relationship between radian measure and degree measure.

      • 4.T1.4 Sketch, in standard position, an angle with a measure expressed in the form kπ radians, where k ∈ Q.

      • 4.T1.5 Express the measure of an angle in radians (exact value or decimal approximation), given its measure in degrees.

      • 4.T1.6 Express the measure of an angle in degrees (exact value or decimal approximation), given its measure in radians.

      • 4.T1.7 Determine the measures, in degrees or radians, of all angles in a given domain that are coterminal with a given angle in standard position.

      • 4.T1.8 Determine the general form of the measures, in degrees or radians, of all angles that are coterminal with a given angle in standard position.

      • 4.T1.9 Explain the relationship between the radian measure of an angle in standard position and the length of the arc cut on a circle of radius r, and solve problems based upon that relationship.

    • 4.T2 Develop and apply the equation of the unit circle.

      • 4.T2.1 Derive the equation of the unit circle from the Pythagorean theorem.

      • 4.T2.2 Generalize the equation of a circle with centre (0,0) and radius r.

      • 4.T2.3 Describe the six trigonometric ratios, using a point P(x, y) that is the intersection of the terminal arm of an angle and the unit circle.

    • 4.T3 Solve problems, using the six trigonometric ratios for angles expressed in degrees and radians.

    • 4.T5 Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.

      • 4.T5.1 Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form when possible.

      • 4.T5.2 Determine, using technology, the approximate solution of a trigonometric equation.

      • 4.T5.3 Verify, with or without technology, that a given value is a solution to a trigonometric equation.

      • 4.T5.4 Identify and correct errors in a solution for a trigonometric equation.

5 Trigonometric Functions and Graphs

  • 5.T Trigonometry

    • 5.T4 Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems.

    • 5.T5 Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.

      • 5.T5.1 Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form when possible.

      • 5.T5.2 Determine, using technology, the approximate solution of a trigonometric equation.

      • 5.T5.3 Verify, with or without technology, that a given value is a solution to a trigonometric equation.

      • 5.T5.5 Relate the general solution of a trigonometric equation to the zeros of the corresponding function (restricted to sine and cosine functions).

6 Trigonometric Identities

  • 6.T Trigonometry

    • 6.T6 Prove trigonometric identities, using:

      • 6.T6.a reciprocal identities

      • 6.T6.b quotient identities

      • 6.T6.c Pythagorean identities

      • 6.T6.d sum or difference identities (restricted to sine, cosine and tangent)

      • 6.T6.e double-angle identities (restricted to sine, cosine and tangent).

      • 6.T6.1 Explain the difference between a trigonometric identity and a trigonometric equation.

      • 6.T6.2 Determine, graphically, the potential validity of a trigonometric identity, using technology.

      • 6.T6.3 Determine the non-permissible values of a trigonometric identity.

      • 6.T6.4 Verify a trigonometric identity numerically for a given value in either degrees or radians.

      • 6.T6.5 Prove, algebraically, that a trigonometric identity is valid.

      • 6.T6.6 Simplify trigonometric expressions using trigonometric identities.

      • 6.T6.7 Determine, using the sum, difference and double-angle identities, the exact value of a trigonometric ratio.

      • 6.T6.8 Explain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid.

    • 6.T5 Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.

      • 6.T5.1 Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form when possible.

      • 6.T5.2 Determine, using technology, the approximate solution of a trigonometric equation.

      • 6.T5.3 Verify, with or without technology, that a given value is a solution to a trigonometric equation.

      • 6.T5.4 Identify and correct errors in a solution for a trigonometric equation.

7 Exponential Functions

8 Logarithmic Functions

9 Permutations, Combinations and the Binomial Theorem

  • 9.PCBT Permutations, Combinations and The Binomial Theorem

    • 9.PCBT1 Apply the Fundamental Counting Principle to solve problems.

      • 9.PCBT1.1 Count the total number of possible choices that can be made, using graphic organizers such as lists and tree diagrams.

      • 9.PCBT1.2 Explain, using examples, why the total number of possible choices is found by multiplying rather than adding the number of ways the individual choices can be made.

      • 9.PCBT1.3 Solve a simple counting problem by applying the Fundamental Counting Principle.

    • 9.PCBT2 Determine the number of permutations of n elements taken r at a time to solve problems.

      • 9.PCBT2.1 Count, using graphic organizers such as lists and tree diagrams, the number of ways of arranging the elements of a set in a row.

      • 9.PCBT2.2 Determine, in factorial notation, the number of permutations of n different elements taken n at a time to solve a problem.

      • 9.PCBT2.3 Determine, using a variety of strategies, the number of permutations of n different elements taken r at a time to solve a problem.

      • 9.PCBT2.4 Explain why n must be greater than or equal to r in the notation ?P?.

      • 9.PCBT2.5 Given a value of k, k ∈ N, solve ?P? = k for either n or r.

      • 9.PCBT2.6 Explain, using examples, the effect on the total number of permutations when two or more elements are identical.

      • 9.PCBT2.7 Solve problems involving permutations with constraints.

    • 9.PCBT3 Determine the number of combinations of n different elements taken r at a time to solve problems.

      • 9.PCBT3.1 Explain, using examples, the differences between a permutation and a combination.

      • 9.PCBT3.2 Determine the number of combinations of n different elements taken r at a time to solve a problem.

      • 9.PCBT3.3 Explain why n must be greater than or equal to r in the notation ?C? or (n choose r).

      • 9.PCBT3.4 Explain, using examples, why ?C? = ?C??? or (n choose r) = (n choose n – r).

      • 9.PCBT3.5 Given a value of k, k ∈ N, solve ?C? = k or (n choose r) = k for either n or r.

    • 9.PCBT4 Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers).