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Learning Outcomes:
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Algebra Review
Note: A review of the following outcomes is suggested, but not required.
It is expected that learners should 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; anduse systems of linear equations to solve applied problems.
Functions and Graphs
It is expected that learners should 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 the domain and range, and the 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, expansions and compressions;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; andfind 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; and decompose a function as a composition of two functions.
Polynomial and Rational Functions
It is expected that learners should 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 a 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; andsolve 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; andgraph a rational function identifying all asymptotes.
Exponential and Logarithmic Functions
It is expected that learners should be able to:
evaluate exponential functions including functions with base e;recognize the inverse relationship between exponential and logarithmic functions;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; anduse 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; and use a graphing utility to solve exponential and logarithmic functions.
Trigonometric Functions
It is expected that learners should be able to:
identify angles in standard position, positive and negative angles, co-terminal 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 a = r È;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° (ð/6), 45° (ð/4), and 60° (ð/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, 2ð); anduse 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(absin C);convert between linear speed and angular speed of an object moving in circular motion using the formula v = rѠ;use the graphing utility to graph trigonometric functions;use half-angle formulas to find exact values; anduse a graphing utility to verify or to approximate the solutions of a trigonometric equation.
Sequences and Series
It is expected that learners should 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; and use sequences and series to model and solve real-life problems.
Optional Learning Outcomes:
use a graphing utility to find the sum of n terms of a sequence.
Optional Topics
Learners may wish to complete any of the following topics but these outcomes are not required:
I. Conic Sections:
a. recognize the equations of the four basic conics: circles, ellipses, hyperbola and parabola;
b. write the standard forms of equations of circles, ellipses, and hyperbola with centre at origin and translated centre (h, k);
c. find the centre and radius of a circle, given its equation, and sketch the graph;
d. find the centre, vertices and foci of an ellipse, given its equation, and sketch the graph;
e. find the centre, vertices, foci and asymptotes of a hyperbola, given its equation, and sketch the graph;
f. find the vertex, focus and directrix of a parabola, given its equation, and sketch the graph;
g. solve nonlinear systems of equations;
h. use nonlinear systems of equations to solve applied problems;
i. use a graphing utility to graph conic sections; and
j. use a graphing utility to solve non-linear systems.
II. Permutations and Combinations:
a. evaluate factorial notation;
b. evaluate permutation and combination notation;
c. solve related applied problems; and
d. use the fundamental counting principle (factorial).
III. Binomial Expansion:
a. expand a power of a binomial using Pascal‘s triangle or factorial notation;
b. find a specific term of a binomial expansion; and
c. find the total number of subsets of a set of n objects.
IV. Probability:
a. compute the probability of a simple event;
b. distinguish between experimental and theoretical probability; and
c. classify events as dependent or independent.
V. Calculus:
a. understand and find the limits of polynomial and rational expressions;
b. find the slope of a line tangent to a curve at a point on the curve;
c. determine the equation of a line tangent to a curve at a given point;
d. use the definition of a derivative to find the derivative of certain polynomials;
e. find derivatives using the power rule;
f. 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; and
g. solve applied maximum/minimum problems.
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