B.C.'s Energy College

## 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
- Imaginary, complex numbers
- 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