Mathematics: Provincial Level—Algebra and Trigonometry
Goal Statement
The goals of the Provincial Algebra and Trigonometry are to prepare adult learners with the knowledge and skills in algebra and trigonometry necessary for entry to technical, vocational and career programs that require Math 12 equivalency as a prerequisite and for future study in higher-level math courses at college/university.
Learning Outcomes
1. Algebra Review
Note: the following outcomes are suggested, but not required. It is expected that learners will be able to:
- recognize subsets and identify properties of real numbers
- use interval notation to write a set of numbers
- evaluate absolute value of a real number and find the distance between two real numbers
- use rules for order of operations and properties of exponents to simplify expressions
- add, subtract, and multiply polynomials and factor a polynomial completely
- determine the domain of a rational expression, simplify rational expressions, perform operations with rational expressions and simplify complex rational expressions
- use properties of exponents to simplify radical expressions
- rationalize the denominator or numerator in a rational expression
- use properties of radicals to simplify and combine radicals
- define imaginary and complex numbers, express them in standard form, and perform operations with complex numbers
- solve linear equations, equations with absolute value, quadratic equations, radical equations, and equations reducible to a quadratic form
- solve linear inequalities, combined inequalities, and absolute value inequalities and graph the solutions on a number line
- solve applied problems using linear and quadratic equations
- solve equations of variation and applied problems involving variation
- solve systems of linear equations in two variables and in three variables
- distinguish between consistent/inconsistent and dependent/independent systems
- use systems of linear equations to solve applied problems
2. Functions and Graphs
It is expected that learners will be able to:
- find the distance between two points in the plane and find the midpoint of a segment
- apply the distance formula and mid-point formula to solve problems
- recognize graphs of common functions: linear, constant, quadratic, cubic, square root, absolute value, reciprocal
- use the vertical line test to identify functions
- graph functions and analyze graphs of functions, identifying: domain and range; intervals on which the function is increasing, decreasing or constant
- write formulas or functions to model real life applications
- determine whether a graph is symmetric with respect to the x-axis, y-axis, and the origin
- identify even or odd functions and recognize their symmetries
- graph transformations of functions: translations, reflections, stretchings and shrinkings
- graph functions defined piecewise
- find the sum, difference, product and quotient of two functions and determine their domains
- find the composition of two functions f and g, finding formulas for f(g(x)) and g(f(x)), identifying the domain of the composition and evaluating the composite function
- given an equation defining a relation, write an equation of the inverse relation
- given a graph of a relation or function, sketch a graph of its inverse
- use the horizontal line test to determine if a function is one-to-one and therefore has an inverse that is a function
- find a formula for the inverse of a function
- find f–1(f(x)) and f(f–1(x)) for any number x in the domains of the functions when the inverse of a function is also a function
Optional Learning Outcomes:
- use a graphing utility to graph functions
- decompose a function as a composition of two functions
3. Polynomial and Rational Functions
It is expected that learners will be able to:
- graph quadratic functions and analyze graphs of quadratic functions identifying the vertex, line of symmetry, maximum/minimum values, and intercepts
- solve applied problems involving maximum and minimum function values
- determine the behaviour of the graphs of polynomial functions of higher degree using the leading coefficient test
- determine whether a function has a real zero between two real numbers
- recognize characteristics of the graphs of polynomial functions including real zeros, y-intercept, relative maxima and minima, domain and range
- divide polynomials using long division
- use synthetic division to divide a polynomial by x – r
- use the remainder and factor theorems to find function values and factors of polynomial
- list the possible rational zeros for a polynomial function with integer coefficients
- factor polynomial functions and find the zeros
- find a polynomial with specified zeros
- solve polynomial and rational inequalities
Optional Learning Outcomes:
- fit a quadratic function to data when three data points are given
- use a graphing utility to graph polynomial functions, determine the real zeros and estimate the relative maxima and minima of a function
- graph a rational function identifying all asymptotes
- determine and analyze complex roots of a polynomial
4. Exponential and Logarithmic Functions
It is expected that learners will be able to:
- evaluate exponential functions including functions with base e
- recognize the inverse relationship between exponential and logarithmic equations
- graph exponential and logarithmic functions including transformations and analyze the graphs in terms of: x- or y-intercepts, asymptotes, increasing or decreasing, domain and range
- convert between exponential and logarithmic equations
- find common and natural logarithms using a calculator
- use basic and inverse properties of logarithms: logb b=1, logb 1=0, logb bx =x, blogbx =x
- use the product rule, quotient rule and power rule to expand or condense logarithmic expressions
- use the change of base property to find a logarithm with base other than 10 or e
- solve exponential and logarithmic equations
- use exponential and logarithmic equations to model and solve real-life applications including exponential growth and decay
Optional Learning Outcomes
- use a graphing utility to graph exponential and logarithmic functions
- use a graphing utility to solve exponential and logarithmic equations
5. Trigonometric Functions
It is expected that learners will be able to:
- identify angles in standard position, positive and negative angles, coterminal angles and reference angles
- convert between degree and radian measures of angles
- find the length of an arc, radian measure of central angle, or radius of a circle using the formula s = r
- identify special angles on a unit circle
- determine the six trigonometric functions of an angle in standard position given a point on its terminal side
- find the exact values of the trigonometric functions of special acute angles 30° (π/6), 45° (π/4), and 60° (π/3) or any angles that are multiples of these special angles
- graph the six trigonometric functions and state their properties
- graph transformations of the sine and cosine functions and determine period, amplitude, and phase shift
- recognize and use the reciprocal, quotient and Pythagorean identities
- apply the sum or difference formulas and double angle formulas to find exact values and to verify trigonometric identities
- recognize and use inverse trigonometric function notation
- use a calculator to evaluate inverse trigonometric functions
- find exact values of composite functions with inverse trigonometric functions
- solve trigonometric equations over the interval (0, 2π)
- use trigonometric functions to model and solve real-life problems
Optional Learning Outcomes
- use the Law of Sines and the Law of Cosines to solve oblique triangles
- solve applied problems using the Law of Sines and the Law of Cosines
- find the area of a triangle given the lengths of any two sides and the measure of the included angle: Area = ½(bcsin A) = ½(ac sin B) = ½(ab sin C)
- convert between linear speed and angular speed of an object moving in circular motion using the formula v = r
- use the graphing utility to graph trigonometric functions
- use half-angle formulas to find exact values
- use a graphing utility to verify or to approximate the solutions of a trigonometric equation
6. Sequences and Series
It is expected that learners will be able to:
- find terms of sequences given the general or nth term
- find a formula for the general or nth term of a given sequence
- use summation notation to write a series and evaluate a series designated in summation notation
- construct the terms of a sequence defined by a recursive formula
- recognize and write terms of arithmetic and geometric sequences
- use nth term formulas for arithmetic and geometric sequences to find a specified term, or to find n when an nth term is given
- find the sum of the first n terms of arithmetic and geometric sequences
- find the sum of an infinite geometric series, if it exists
- use sequences and series to model and solve real-life problem
Optional Learning Outcomes:
- use a graphing utility to find the sum of n terms of a sequence
7. Optional Topics
Learners may wish to complete any of the following topics but these outcomes are not required:
A. Conic Sections
- recognize the equations of the four basic conics: circles, ellipses, hyperbola and parabola
- write the standard forms of equations of circles, ellipses, and hyperbola with centre at origin and translated centre (h, k)
- find the centre and radius of a circle, given its equation, and sketch the graph
- find the centre, vertices and foci of an ellipse, given its equation, and sketch the graph
- find the centre, vertices, foci and asymptotes of a hyperbola, given its equation, and sketch the graph
- find the vertex, focus and directrix of a parabola, given its equation, and sketch the graph
- solve nonlinear systems of equations
- use nonlinear systems of equations to solve applied problems
- use a graphing utility to graph conic sections
- use a graphing utility to solve nonlinear systems
B. Permutations and Combinations
- evaluate factorial notation
- evaluate permutation and combination notation
- solve related applied problems
- use the fundamental counting principle (factorial)
C. Binomial Expansion
- expand a power of a binomial using Pascal’s triangle or factorial notation
- find a specific term of a binomial expansion
- find the total number of subsets of a set of n objects
D. Probability
- compute the probability of a simple event
- distinguish between experimental and theoretical probability
- classify events as dependent or independent
E. Calculus
- understand and find the limits of polynomial and rational expressions
- find the slope of a line tangent to a curve at a point on the curve
- determine the equation of a line tangent to a curve at a given point
- use the definition of a derivative to find the derivative of certain polynomials
- find derivatives using the power rule
- use the derivative to graph and analyze functions in terms of: increasing/decreasing intervals, minimum/maximum points, concave up/concave down intervals, and inflection points
- solve applied maximum/minimum problems