45: Amplitude, Period, Phase Shift, and Vertical Translation
46: Trigonometric Identities
47: Sum Formulas
48: The Sine and Cosine Laws
51: Introduction To The Circle
52: Circle Equation
53: Circle and Chords
54: Circles, Tangents, and Problem Solving
55: The Parabola
56: The Parabola - Vertex Not at the Origin
57: The Ellipse - Centre at the Origin
58: The Hyperbola - Centre at the Origin
59: Quadratic Relations and Conic Sections
61: The Normal Distribution
62: Bivariate Data
63: Confidence
71 Permutations
72: Combinations
Module 11: Introduction to Polynomial Functions
Objective
1. Recognize a polynomial function.
2. Classify a given polynomial function according to degree and coefficients (integral, rational, or real).
3. Divide an integral polynomial in one variable by a binomial of the form (x – a) using the division algorithm.
4. Divide an integral polynomial in one variable by a binomial of the form (x – a) using synthetic division.
Module 12: The Remainder Theorem and Factor Theorem
Objective
1. Evaluate an integral polynomial function for a given value of the domain by substitution.
2. Develop and prove the Remainder Theorem.
3. Evaluate an integral polynomial function for a given value of the domain using the Remainder Theorem.
4. Develop and prove the Factor Theorem.
Module 13: Factoring and Zeros
Objective
1. Develop integral and Rational Zero Theorems.
2. Factor a given integral polynomial over the integers.
3. Apply the factor theorem to solve equations involving polynomials of degree greater than 2.
4. Determine an integral polynomial function given its rational zeros.
Module 14: Graphing
Objective
1. Sketch the graph of a given polynomial function.
2. Determine a polynomial function that defines a given graph.
3. Determine the possible sums of the multiplicities of the zeros of a given real polynomial function.
4. Identify all zeros of a given real polynomial function.
Module 21: Exponential Functions
Objective
1. Maintain previous skills with exponential expressions.
2. Identify and graph exponential functions.
3. Solve exponential equations by transforming both sides to a common base.
4. Write and solve exponential functions describing problems involving exponential growth and decay.
Module 22: Introduction To Logarithms
Objective
1. Define the logarithmic functions.
2. Identify and graph logarithmic functions.
3. Transform an equation in exponential form to logarithmic form and vice versa.
4. Solve equations of the form y = loga x, a > 0, a is not equal to 1 using exponential form.
Module 23: Laws of Logarithms
Ojective
1. Derive the laws of logarithms from the exponent laws.
2. Apply the laws of logarithms in simplifying or evaluating expressions involving logarithms.
Module 24: Applications
Objective.
1. Solve exponential equations.
2. Solve logarithmic equations.
3. Apply exponential and logarithmic functions to solve problems.
Module 31: Sequences
Objective
1. Identify and use the following: sequences, terms of a sequence, finite sequences, and infinite sequences.
2. Determine the terms of a sequence given its defining rule.
3. Investigate patterns in sequences.
4. Determine the terms of a sequence that is defined recursively.
5. Determine a recursive formula of a sequence given the first three terms of the sequence.
Module 32: Arithmetic Sequences
Objective
1. Identify an arithmetic sequence.
2. Determine the common difference of a given arithmetic sequence.
3. Derive the general term formula for an arithmetic sequence.
4. Apply the general term formula to determine the first term, the nth term, the common difference, or the number of terms for a given arithmetic sequence.
5. Solve problems involving arithmetic sequences.
Module 33: Geometric Sequences
Objective
1. Recognize a geometric sequence.
2. Determine the common ratio (r) of a given geometric sequence.
3. Derive the general term formula for a geometric sequence.
4. Apply the general formula to determine the first term, the last term, the common ratio, or the number of terms for a given sequence.
5. Derive and apply the formula A = P(1 + i)n.
6. Solve problems involving geometric sequences.
Module 34: Series
Objective
1. Identify and use the following: series, term of a series, infinite series, and finite series.
2. Expand a series given in sigma notation.
3. Express a given series in sigma notation.
4. Determine the sum of a given finite series.
Module 35: Arithmetic Series
Objective
1. Identify an arithmetic series.
2. Derive the sum formulas for an arithmetic series.
3. Apply a sum formula to determine the number of terms, the first term, the last term, or the sum of terms for a given arithmetic series.
4. Solve problems involving a finite arithmetic series.
Module 36: Geometric Series
Objective
1. Identify a geometric series.
2. Derive the sum formula for a geometric series.
3. Apply a sum formula to determine the number of terms, the first term, or the sum of the terms for a given geometric series.
4. Solve problems involving a finite geometric series.
5. Apply concepts of geometric series to solve problems involving annuities.
Module 37 Infinite Series and Limits
Objective
1. Identify an infinite sequence as convergent or divergent.
2. Determine the limit of an infinite convergent sequence.
3. Determine the limit of a given function as n approaches infinity.
4. Determine the sum of an infinite geometric series (|r|<1) by finding the limit of its sequence of partial sums.
5. Derive the sum formula of an infinite geometric series.
6. Solve problems involving the application of , for |r|<1.
Module 41: Circular Paths
Objective
1. Define and identify the values of the trigonometric ratios for angles in standard position in terms of x, y, and r.
2. Derive the values of any trigonometric ratio given the value of one of the others.
3. Determine the reference angle of any given angle and define the value of a trigonometric ratio for an angle in terms of the reference angle.
Module 42: The Unit Circle
Objective
1. Relate the trigonometric functions to coordinates of points on a unit circle.
2. Define and use radian measure.
3. Examine the relationship between radian and degree measure and determine trigonometric function values of angles measured in radians.
4. Determine and use the exact relative measures of the sides of 30˚ – 60˚ – 90˚ triangles and 45˚ – 45˚ – 90˚ triangles and derive the exact value of the trigonometric ratios for angles with radian measure , where n is an integer.
Module 43: Solving Equations
Objective
1. Solve first-degree trigonometric equations.
2. Solve second-degree trigonometric equations.
3. Solve trigonometric equations involving multiples of angles.
Module 44: Graphs of Trigonometric Functions
Objective
1. Graph the six trigonometric functions and identify their domain, range, and period.
Module 45: Amplitude, Period, Phase Shift, and Vertical Translation
Objective
1. Identify the amplitude, period, phase shift, vertical translation, and range of the graphs of functions of the form y = a sin b(q + c) + d or y = a cos b(q + c) + d.
2. Identify and produce the graphs of functions of the form y = a sin b(q + c) + d or y = a cos b(q + c) + d.
Module 46: Trigonometric Identities
Objective
1. Derive and apply the quotient relations, reciprocal relations, or Pythagorean relations to expand or simplify the trigonometric expressions.
Module 47: Sum Formulas
Objective
1. Derive and apply the negative angle relations and complementary angle relations.
2. Derive and apply the sum and difference formulas.
Module 48: The Sine and Cosine Laws
Objective
1. Develop and apply the Sine Law.
2. Develop and apply the Cosine Law.
3. Solve problems involving the ambiguous case of the Law of Sines.
4. Solve problems involving regular polygons.
Module 51: Introduction To The Circle
Objective
1. Identify and use the terms: circle, centre, radius, diameter, point on the circle, chord, tangent, point of tangency, general form, standard form.
2. Derive the equation of the circle in standard form from a locus definition.
3. Convert the equation of the circle from standard form to general form and vice versa.
4. Determine the components of a circle given the equation.
5. Draw the graph of a circle, given the centre and the radius or the equation of the form (x – h)2 + (y – k)2 = r2.
6. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given the coordinates of the centre (h, k) and radius (r).
Module 52: Circle Equation
Objective
1. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given the coordinates of the centre and a point on the circle.
2. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given the end points of a diameter of the circle.
Module 53: Circle and Chords
Objective
1. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given two points on the circle and the equation of the line containing the centre of the circle.
2. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given the coordinates of three points on the circle.
Module 54: Circles, Tangents, and Problem Solving
Objective
1. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given the coordinates of the centre and the equation of a line tangent to the circle.
2. Determine the equation of a circle in the form (x – h)2 + (y – k)2 = r2, given the equation of a line tangent to the circle, point of tangency and an equation of a line containing the centre.
3. Solve problems related to the circle.
Module 55: The Parabola
Objective
1. Identify and use the terms: parabola, focus, vertex, axis of symmetry, directrix, and point on the parabola.
2. Derive the standard form of the equation of a parbola with vertex at the origin from the locus definition.
3. Determine the focus and the directrix given the equation of a parabola with vertex at the origin.
4. Draw the graph of a parabola with vertex at the origin given the focus, the directrix, or the equation.
5. Determine the equation of a parabola with vertex at the origin given the coordinates of the focus or the equation of the directrix.
6. Determine the equation of a parabola with vertex at the origin given one point on the parabola and its focus, directrix, or axis of symmetry.
7. Solve problems involving parabolas with vertex at the origin.
Module 56: The Parabola - Vertex Not at the Origin
Objective
1. Derive the standard form of the equation of a parabola with the vertex not at the origin from the locus definition.
2. Determine the focus and the directrix given the equation of a parabola with vertex not at the origin.
3. Convert the equation of a parabola from standard form to general form and vice versa.
4. Draw the graph of a parabola with vertex not at the origin given its focus, directrix, or equation.
5. Determine the equation of a parabola with vertex not at the origin given a combination of two of the following: coordinates of vertex, coordinates of the focus, equation of the directrix.
6. Determine the equation of a parabola with vertex not at the origin given one point on the parabola, the coordinates of the vertex, and the orientation of the axis of symmetry.
7. Solve problems related to parabolas with vertex not at the origin.
Module 57: The Ellipse - Centre at the Origin
Objective
1. Identify and use the terms: ellipse, centre, focus, major axis, minor axis, vertex, and focal radii.
2. Derive the standard form of the equation of an ellipse with centre at the origin from a locus definition.
3. Determine the components of an ellipse given its equation.
4. Convert the equation of an ellipse from standard form to general form and vice versa.
5. Draw the graph of an ellipse with centre at the origin given its equation or some of its components.
6. Determine the equation of an ellipse given components of the ellipse from which a and b may be calculated.
7. Determine the equation of an ellipse given one point and one component of the ellipse.
8. Solve problems involving ellipses.
Module 58: The Hyperbola - Centre at the Origin
Objective
1. Identify and use the terms: hyperbola, centre, focus, vertex, transverse axis, conjugate axis, asymptotes, and focal radii.
2. Derive the standard form of the equation of a hyperbola from a locus definition.
3. Determine the components of a hyperbola given its equation.
4. Convert the equation of a hyperbola from standard form to general form and vice versa.
5. Draw the graph of a hyperbola with centre at the origin given its equation or some of its components.
6. Determine the equation of a hyperbola given components of the hyperbola.
7. Determine the equation of a hyperbola given one point and one component of the hyperbola.
8. Solve problems involving hyperbolas.
Module 59: Quadratic Relations and Conic Sections
Objective
1. Define circle, parabola, ellipse and hyperbola as conic sections.
2. Distinguish between general forms of the equations of circles, parabolas, ellipses and hyperbolas.
3. Solve problems involving the circle, parabola, ellipse and/or hyperbola.
Module 61: The Normal Distribution
Objective
1. Calculate and interpret the mean and standard deviation of a set of data.
2. Identify the normal distribution.
3. Use z-scores to solve situations that are normally distributed.
4. Use z-scores to calculate the probability of an event happening.
Module 62: Bivariate Data
Objective
1. Plot sets of bivariate data to produce a scatterplot.
2. Plot a line of best fit on a scatterplot using the median fit method.
3. Use the equation of the line of best fit to generate new data for a population.
4. Determine the strength and type of correlation between the variables of a bivariate distribution.
5. Collect, organize, and analyze sets of bivariate data.
Module 63: Confidence
1. Design and administer a yes/no simple survey and collect and organize the results of the survey.
2. Draw box and whisker plots of the results of multiple samples.
3. Use a 90% box and whisker plot chart to find the confidence interval for a survey result.
4. Draw statistical conclusions and make inferences to populations, and explain the confidence with which such conclusions and inferences are made based on the results of yes/no surveys.
5. Assess the strength, weaknesses, and biases of given samples.
Module 71 Permutations
1. Solve problems using the Fundamental Counting Principle.
2. Define and use nPn and n!
3. Define and use nPr; also solve linear permutations with repetitions and permutations with like elements.
4. Solve problems involving circular permutations and permutations with reflections.
5. Solve probability questions which involve the use of permutations.
Module 72: Combinations
Objective
1. Define the meaning and use applications of the formula nCr .
2. Solve probability questions which involve the use of combinations.
3. Define and use the Binomial Theorem and Pascal's Triangle.
