Newfoundland and Labrador

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Skills available for Newfoundland and Labrador grade 12 math curriculum

Objectives are in black and IXL math skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practise that skill.

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Polynomial Functions

  • Develop algebraic and graphical reasoning through the study of relations.

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

      • 12.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.

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

      • 12.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).

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

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

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

      • 12.RF11.1 Identify the polynomial functions in a set of functions, and explain the reasoning.

      • 12.RF11.2 Explain the role of the constant term and leading coefficient in the equation of a polynomial function with respect to the graph of the function.

      • 12.RF11.3 Generalize rules for graphing polynomial functions of odd or even degree.

      • 12.RF11.4 Explain the relationship among the following:

      • 12.RF11.5 Explain how the multiplicity of a zero of a polynomial function affects the graph.

      • 12.RF11.6 Sketch, with or without technology, the graph of a polynomial function.

      • 12.RF11.7 Solve a problem by modeling a given situation with a polynomial function.

      • 12.RF11.8 Determine the equation of a polynomial function given its graph.

Function Transformations

  • Develop algebraic and graphical reasoning through the study of relations.

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

      • 12.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.

      • 12.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.

      • 12.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.

      • 12.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.

      • 12.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).

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

      • 12.RF4.A x-axis

      • 12.RF4.B y-axis

      • 12.RF4.C line y = x.

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

      • 12.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.

      • 12.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.

      • 12.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.

      • 12.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 xaxis or the y-axis.

      • 12.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.

      • 12.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.

      • 12.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.

      • 12.RF4.9 Write the equation of a function, given its graph which is a reflection of the graph of the function y = f(x) through the line y = x.

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

      • 12.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.

      • 12.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.

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

      • 12.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.

      • 12.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).

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

      • 12.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.

      • 12.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).

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

Radical Functions

  • Develop algebraic and graphical reasoning through the study of relations.

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

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

      • 12.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.

      • 12.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.

      • 12.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.

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

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

Trigonometry and the Unit Circle

  • Develop trigonometric reasoning.

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

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

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

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

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

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

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

      • 12.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.

      • 12.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.

      • 12.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.

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

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

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

      • 12.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.

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

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

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

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

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

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

Trigonometric Functions and Graphs

  • Develop trigonometric reasoning.

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

      • 12.T4.1 Sketch, with or without technology, the graphs of y = sin x and y = cos x.

      • 12.T4.2 Determine the characteristics (amplitude, domain, period, range and zeros) of the graphs of y = sin x and y = cos x.

      • 12.T4.3 Determine how varying the value of a affects the graph of y = a sin x and y = a cos x.

      • 12.T4.4 Determine how varying the value of b affects the graphs of y = sin bx and y = cos bx.

      • 12.T4.5 Determine how varying the value of d affects the graphs of y = sin x + d and y = cos x + d.

      • 12.T4.6 Determine how varying the value of c affects the graphs of y = sin(x + c) and y = cos(x + c).

      • 12.T4.7 Sketch, without technology, graphs of the form y = a sinb (x – c)+ d and y = a cosb (x – c)+ d using transformations, and explain the strategies.

      • 12.T4.8 Determine the characteristics (amplitude, domain, period, phase shift, range and zeros) of the graphs of trigonometric functions of the form y = a sinb (x – c)+ d and y = a cosb (x – c)+ d.

      • 12.T4.9 Determine the values of a, b, c and d for functions of the form y = a sin b(x - c) + d and y = a cos b(x - c) + d that correspond to a given graph, and write the equation of the function.

      • 12.T4.10 Solve a given problem by analyzing the graph of a trigonometric function.

      • 12.T4.11 Explain how the characteristics of the graph of a trigonometric function relate to the conditions in a problem situation.

      • 12.T4.12 Determine a trigonometric function that models a situation to solve a problem.

      • 12.T4.13 Sketch, with or without technology, the graph of y = tan x.

      • 12.T4.14 Determine the characteristics (asymptotes, domain, period, range and zeros) of the graph of y = tan x.

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

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

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

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

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

Trigonometric Identities

  • Develop trigonometric reasoning.

    • 12.T6 Prove trigonometric identities, using:

      • 12.T6.A reciprocal identities

      • 12.T6.B quotient identities

      • 12.T6.C Pythagorean identities

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Exponential Functions

Logarithmic Functions

Permutations, Combinations and Binomial Theorem

  • Develop algebraic and numeric reasoning that involves combinatorics.

    • 12.PCBT1 Apply the fundamental counting principle to solve problems.

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

      • 12.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.

      • 12.PCBT1.3 Solve a simple counting problem by applying the fundamental counting principle.

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

      • 12.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.

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

      • 12.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.

      • 12.PCBT2.4 Explain why n must be greater than or equal to r in the notation subscript n Pr.

      • 12.PCBT2.5 Given a value of k, k ε N, solve subscript n Pr = k for either n or r.

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

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

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

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

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

      • 12.PCBT3.3 Explain why n must be greater than or equal to r in the notation subscript n Cr or binomial coefficient (n r).

      • 12.PCBT3.4 Explain, using examples, why subscript n Cr = subscript n C subscript (n-r) or binomial coefficient (n r) = binomial coefficient (n n-r).

      • 12.PCBT3.5 Given a value of k, k ε N, solve subscript n Cr = k or binomial coefficient (n r) = k for either n or r.

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