## MATH 050 - Provincial Algebra and Trigonometry

In Provincial Level Mathematics, students study the following types of functions: polynomial, quadratic, logarithmic, exponential, exponential, and trigonometric. This course prepares the adult learners with the necessary skills and knowledge for entry into technical, vocational, and career programs that require Math 12 equivalency as a prerequisite and for future study in higher-level math course at College/University.

**Hours:** 120 (Lecture Hours: 6)

**Total Weeks:** 20

**Prerequisites:**

75% in Math 040 Advanced Algebraic mathematics, Math 11 or equivalent,

OR permission of the instructor

**Non-Course Prerequisites:**

None

**Co-Requisites:**

None

**Course Content:**

Algebra Review

- Subsets and properties of real numbers

- Absolute value, orders of operations, properties of exponents

- Addition, subtraction, multiplication, division and factor polynomials

- Rational expressions, radical expressions

- Imaginary, complex numbers

- Linear, quadratic, radical, and reducible equations

- Linear, combined, absolute value inequalities

- Equations of variation and applied problems

- Systems of linear equations in two and three variables

Functions and Graphs

- Two points in a plane and midpoint of a segment

- Distance and midpoint formulas

- Graphs of common functions: linear, constant, quadratic, cubic, square root, absolute value, reciprocal

- Vertical line test

- Domain, range, intervals of increase, decrease, constant for graphs and graph functions

- Real life applications formulas and functions

- Symmetry of x- and y-axes, odd or even functions

- Translation, reflection, stretching, and shrinking of graph transformation of functions

- Sum, difference, product, and quotient of two functions

- Two functions, f and g finding formulas for f(g(x)) and g(f(x)), domain of and composite function

- Equation defining a relation and equation of the reverse relation

- Graph of a relation and graph of the reverse

- Horizontal line test to determine if function is one-to-one and therefore has a reverse

- Formula for the reverse of a function

- f-1(f(x)) and f(f-1(x))for any number x in the domains of the functions when the reverse of a function is also a function

Polynomial and Rational Functions

- Quadratic functions and analysis of function for vertex, line of symmetry, maximum/minimum values, intercepts

- Applied problems involving maximum and minimum function values

- Graphs of polynomial functions of higher degrees using the leading coefficient test

- Real zeros between two real numbers of a function

- Graphs of polynomial functions, including real zeros, y-intercept, relative maxima and minima, domain and range

- Long division of polynomials

- Synthetic division of a polynomial by x - r

- Remainder and factor theorems of polynomials

- Rational zeros for a polynomial function with integer coefficients

- Polynomial functions, finding zeros, specified zeros

- Polynomial and rational inequalities

Exponential and Logarithmic Functions

- Exponential functions including functions with base e

- Inverse relationship between exponential and logarithmic functions

- Graphs of exponential and logarithmic functions

- Conversions between exponential and logarithmic functions

- Common and natural logarithms using calculator

- Basic and inverse properties of logarithms

- Product rule, quotient rule, and power rule for expansion and condensation of logarithmic expressions

- Change of base property

- Exponential and logarithmic equations

- Real-life applications of exponential growth and decay

Trigonometric Functions

- Angles in standard position, positive and negative angles, co-terminal and reference angles

- Conversion between between degree and radian measures of angles

- Arc length, radian measure of central angle, radius of a circle with formula s= r 0

- Special angles on a unit circle

- Trigonometric functions of an angle in standard position

- Exact values of trigonometric functions of special acute angles

- Graphing six trigonometric and their properties

-Transformation of the sine and cosine functions- period, amplitude and phase shift

- Reciprocal, quotient, and Pythagorean identities

- Sum or difference formulas, and double angle formula

- Inverse trigonometric function notation and use of calculator

- Composite functions with inverse trigonometric functions

- Trigonometric equations

- Real-life problems and trigonometric functions

Options - Law of Sines and Cosines and oblique triangles

Sequences and Series

- Terms of sequences given the general term or nth term

- Formula for the general or nth term given a sequence

- Summation notation and series evaluation

- Terms of a sequence defined by a recursive formula

- Arithmetic and geometric sequences

- nth term formulas to find a specified term

- The sum of first n terms

- Sum of an infinite geometric series

- Sequences and series to solve real-life problems,

Optional Topics

- Conic sections

- Permutations and combinations

- Binomial expansions

- Probability

- Calculus

**Learning Outcomes:**

Upon successful completion of this course, students will be able to:

Functions and Graphs

- Find the distance between two points in the plane and the midpoint of a segment

- Apply the distance and midpoint 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 and analyze functions, identifying: domain, range, intervals on which the function is decreasing, increasing or constant

- Write formulas or functions to model real-life applications

- Determine graph or function symmetry with respect to the x-axis, y-axis, and origin

- Identify even or odd functions and recognize their symmetry

- Graph transformations, translations, reflections, stretchings, and shrinkings of functions

- Graph functions defined piecewise

- Find the sum of, difference, product, 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))

- Write an equation of the inverse relation given an equation defining the relation

- Sketch a graph of its reverse given the graph of the relation or function

- Use the horizontal line test to determine if a function is one-to-one and therefore has an inverse

- Find a formula for the inverse of a function

- Evaluate composite functions

Polynomial and Rational Functions

- Graph and analyze quadratic functions identifying the vertex, line of symmetry, minimum/maximum values and intercepts.

- Solve applied problems involving minimum and maximum function values

- Determine the behaviour of graphs of polynomial functions of higher degree using the leading coefficient test

- Determine whether a function has a real zero between two real numbers

- Write and manipulate complex numbers

- Divide polynomials using long and synthetic division

- Demonstrate the use of remainder and factor theorems

- Factor polynomial expressions and solve polynomial functions and find the zeros

- Find a polynomial equation given its roots

Exponential and Logarithmic Functions

- Understand the relationship between exponential and logarithmic functions

- Recognize the inverse relationships

- Graph and analyze exponential and logarithmic functions

- Use the laws of exponents and the laws of logarithms to simplify expressions and solve equations

- Use exponential and logarithmic equations to solve real-life applications including exponential growth and decay

Trigonometric Functions

- Identify angles in standard position, positive and negative angles, co-terminal and reference angles

- Identify special angles and use the unit circle and convert between radians and degrees

- Determine the trig function values of an angle in standard position given a point on a terminal arm

- Use trig identities and algebra to simply expressions and solve trig equations

- Graph and analyze the sine, cosine, and tangent functions

- Use a calculator to evaluate inverse trig relations

- Use trig functions to model and solve real-life problems

Series and Sequences

- Distinguish between and solve problems involving arithmetic and geometric sequences and series

- Use the formulas to find terms, positions of terms, arithmetic and geometric means, differences or ratios, sums of series , and sums of series and sums of infinite series.

- Use sequences and series to model and solve real-life problems

**Knowledge:**

Learners will acquire the knowledge, skills and strategies required to analyze, manipulate, graph and interpret a variety of mathematical functions

**Grading System:** Letters

**Passing Grade:** D (50%)

**Percentage of Individual Work:** 100

**Text Books:**

Textbooks are subject to change. Please contact the bookstore at your local campus for current book lists.