Secondary Mathematics - Fourth Year Advanced Algebra, Trigonometry and StatisticsFunctions 1: Functions Generally - Demonstrate knowledge and skill related to functions in general
- Define a function.
- Differentiate a function from a mere relation.
- real life relationships
- set of ordered pairs
- graph of a given set of ordered pairs
- vertical line test
- given equation
- Illustrate the meaning of the functional notation f(x); determine the value of f(x) given a value for x.
Linear Functions 2: Linear Functions and their Graphs - Demonstrate knowledge and skill related to linear functions and apply in solving problems
- Define the linear function f(x) = mx + b.
- Given a linear function Ax + By = C, rewrite in the form f(x) = mx + b and vice versa.
- Draw the graph of a linear function, given:
- any two points
- x and y intercepts
- slope and one point
- slope and the y-intercept
- Given f(x) = mx + b, determine the following attributes of the function:
- slope
- trend: increasing or decreasing
- x and y intercepts
- some points
- Determine f(x) = mx + b given any two parameters.
- slope and y-intercept
- x and y intercepts
- slope and one point
- any two points
- Solve problems involving linear functions.
Quadratic Functions 3: Quadratic Functions and their Graphs - Quadratic functions - an introduction
- Identify quadratic functions f(x) = ax2 + bx + c
- Rewrite a quadratic function ax2 + bx + c in the form f(x) = a(x - h)2 + k and vice versa
- Given a quadratic function, determine the following characteristics of its graph.
- highest or lowest point (vertex)
- axis of symmetry
- direction of opening of the graph
- Given particular characteristics, draw the graph of a quadratic function, showing:
- vertex
- axis of symmetry
- direction of opening of the graph
- given points
- Analyze the effects on the graph of changes in a, h and k in f(x) = a(x-h)2 + k
4: Properties of Quadratic Functions. - Determine the 'zeros of a quadratic function' by relating this to 'roots of a quadratic equation'.
- Find the roots of a quadratic equation,
- by factoring
- using the quadratic formula
- by completing the square
- Derive a quadratic function from:
- zeros of the function
- table of values
- graph
- Solve problems involving quadratic functions and equations.
Polynomial Functions 5: Getting to know Polynomial Functions - Identify a polynomial function from a given set of relations
- Determine the degree of a given polynomial function
- Find the quotient of polynomials
- by algorithm
- by synthetic division
6: Theorems, Zeros and Graphs - Find by synthetic division the quotient and the remainder when p(x) is divided by (x-c)
- State and illustrate the Remainder Theorem
- Find the value of p(x) for x = k.
- synthetic division
- Remainder Theorem
- State and illustrate the Factor Theorem
- Find the zeros of polynomial functions of degree greater than 2
- Factor Theorem
- factoring
- synthetic division
- depressed equations
- Draw the graph of polynomial functions of degree greater than 2 (use graphing calculator if available)
Exponential and Logarithmic Functions 7: Exponential Functions and their Graphs - Identify certain relationships in real life which are exponential (e.g. population growth over time, growth of bacteria over time, etc.)
- Given a table of ordered pairs, state whether the trend is exponential or not
- Draw the graph of an exponential function
- Describe some properties of an exponential function from its graph
- Given the graph of an exponential function determine the domain, range, intercepts, trend and asymptote
- Describe the behavior of the graph of an exponential function
8: Zeros and the Inverse - Use the laws on exponents to find the zeros of exponential functions
- Define inverse functions
- Determine the inverse of a given function
9: Logarithmic Functions and their Graphs - define the logarithmic function as the inverse of the exponential function
- f(x) = logax as the inverse of the exponential function f(x) = ax
- draw the graph of a logarithmic function
- Describe some properties of the logarithmic function from its graph
- State and apply the laws for logarithms
- Solve simple logarithmic equations
- solve problems involving exponential and logarithmic functions (e.g. - exponential growth or decay)
Circular Functions and Trigonometry 10: The Unit Circle - Define a unit circle; arc lengths; unit measures of an angle
- Convert from degree to radian and vice versa
- Illustrate angles in standard position (i.e. initial side coincident with the positive x-axis; coterminal angles; reference angles
- Visualize rotations along the unit circle and relate these to angle measures (clockwise or counterclockwise directions)
- length of an arc
- angles beyond 360o or 2â radians
- Given an angle in standard position in a unit circle, determine the coordinates of its terminal side
- when one coordinate is given (apply the Pythagorean Theorem and the properties of special right triangles)
- when the angle is of the form:
- 1800n ± 300
- 1800n ± 600
- 1800n ± 450
- 900n
11: Circular Functions - Define the six circular functions
- sine
- cosine
- tangent
- cotangent
- secant
- cosecant
- Find the six circular functions of angles with special values
- Draw the graph of the sine, cosine and tangent functions.
- For the angles, - 2â<x<2â or - 360o<x<360o
- Describe the properties of the graphs of the sine, cosine and tangent functions
12: Six Trigonometric Functions and Identities - Define the six trigonometric functions of an angle in standard position
- Find the values of six trigonometric functions of an angle x, given some conditions
- Solve simple trigonometric equations
- State the fundamental trigonometric identities
13: Triangle Trigonometry - Demonstrate ability to apply trigonometric functions, laws of sine and cosine to solve problems
- right triangles
- triangles using the Law of Sines
- triangles using the Law of Cosines
Statistics 14: Basic Statistics - Define the terms statistics, sample, population
- Give the history and importance of the study of statistics
- Use the rules of summation to find sums
- State and explain the different sampling techniques
- Collect statistical data and organize in a table
- Construct frequency distribution tables
- Find the measures of central tendency using ungrouped data
- Find the measures of central tendency using grouped data
15: Measures of Variability - Calculate the different measures of variability relative to a given set of data, grouped or ungrouped
- Give the characteristics of a set of data using the measures of variability
- From a given statistical data, analyze, interpret, draw conclusions, make predictions and make recommendations/decisions.
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