Education eWorld - Grade 11 Functions, University Preparation

Grade 11 Functions, University Preparation

A. Characteristics of Functions

A1. Demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations

Explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., identifying a one-to-one or many-to-one mapping; using the vertical-line test)
Represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions [e.g., evaluate f(½), given f(x) = 2x² + 3x – 1]
Explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions f (x) = x, f (x) = x², f(x) = the square root of x, and f(x) = 1/x; describe the domain and range of a function appropriately (e.g., for y = x² +1, the domain is the set of real numbers, and the range is y is greater than or equal to 1); and explain any restrictions on the domain and range in contexts arising from realworld applications
Relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations)
Determine the numeric or graphical representation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the graph of a function and the graph of its inverse (e.g., the graph of the inverse is the reflection of the graph of the function in the line y = x)
Determine, through investigation, the relationship between the domain and range of a function and the domain and range of the inverse relation, and determine whether or not the inverse relation is a function
Detrmine, using function notation when appropriate, the algebraic representation of the inverse of a linear or quadratic function, given the algebraic representation of the function [e.g., f (x) = (x – 2)² – 5], and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the algebraic representations of a function and its inverse (e.g., the inverse of a linear function involves applying the inverse operations in the reverse order)
Determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x – d)) + c, and describe these roles in terms of transformations on the graphs of f(x) = x, f(x) = x , f(x) = the square root of x, and f(x) = 1/x (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
Sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) = x, f(x) = x², f(x) = the square root of x, and f(x) = 1/x, and state the domain and range of the transformed functions

A2. Determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications

Determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant)
Determine the maximum or minimum value of a quadratic function whose equation is given in the form f (x) = ax2 + bx+ c, using an algebraic method (e.g., completing the square; factoring to determine the zeros and averaging the zeros)
Solve problems involving quadratic functions arising from real-world applications and represented using function notation
Determine, through investigation, the transformational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding quadratic equation and a point on the function
Solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determining the time when two identical cylindrical water tanks contain equal volumes of water, if one tank is being filled at a constant rate and the other is being emptied through a hole in the bottom)

A3. Demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions

Simplify polynomial expressions by adding, subtracting, and multiplying
Verify, through investigation with and without technology, that (the square root of ab) = (the square root of a) x (the square root of b), a is greater than or equal to 0, b is greater than or equal to 0, and use this relationship to simplify radicals (e.g., the square root of 24) and radical expressions obtained by adding, subtracting, and multiplying [e.g., (2 + the square root of 6)(3 – the square root of 12)]
Simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values
Determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values)Exponential Functions

A4. Evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways

Graph, with and without technology, an exponential relation, given its equation in the form y = a to the x power (a > 0, a is not equal to 1), define this relation as the function f(x) = a to the x power, and explain why it is a function
Determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with a rational exponent (i.e., x to the (m/n) power, where x > 0 and m and n are integers)
Simplify algebraic expressions containing integer and rational exponents [e.g., (x³) ÷ (x to the ½ power), ((x to the 6th power)y³) to the 1/3 power], and evaluate numeric expressions containing integer and rational exponents and rational bases [e.g., 2 to the –3 power, (–6)³, 4 to the ½ power, 1.01 to the 120th power]
Determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form f(x) = a to the x power (a > 0, a is not equal to 1), function machines]

A5. Make connections between the numeric, graphical, and algebraic representations of exponential functions

Distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations)
Determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x - d)) + c, and describe these roles in terms of transformations on the graph of f(x) = a to the x power (a > 0, a is not equal to 1) (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
Sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graph of f(x) = a to the x power (a > 0, a is not equal to 1), and state the domain and range of the transformed functions
Determine, through investigation using technology, that the equation of a given exponential function can be expressed using different bases [e.g., f(x) = 9 to the x power can be expressed as f(x) = 3 to the 2x power], and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws)
Represent an exponential function with an equation, given its graph or its properties

A6. Identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications

Collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, ESTAT), and graph the data
Identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)
Solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations

B. Discrete Functions

B1. Demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal's triangle

Make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function [e.g., f(x) = 2x, where the domain is the set of natural numbers, is a discrete linear function, and its graph is a set of equally spaced points; f(x) = 2x, where the domain is the set of real numbers, is a continuous linear function and its graph is a straight line]
Determine and describe (e.g., in words; using flow charts) a recursive procedure for generating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, …), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)
Connect the formula for the nth term of a sequence to the representation in function notation, and write terms of a sequence given one of these representations or a recursion formula
Represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8,16, 32, 64, … as t base 1 = 2; t base n = 2(t base (n–1), as f(n) = 2 to the n power, or as t base n = 2 to the n power, or represent ½, 2/3, 3/4, 4/5, 5/6, 6/7,... as t base 1 = ½; t base n = t base (n-1) + (1/(n(n + 1)), as f(n) = n/(n+1), or as t base n = n/(n + 1), where n is a natural number], and describe the information that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms)
Determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal's triangle, and represent the patterns in a variety of ways (e.g., tables of values, algebraic notation)
Determine, through investigation, and describe the relationship between Pascal's triangle and the expansion of binomials, and apply the relationship to expand binomials raised to whole-number exponents [e.g., (1 + x) to the 4th power, (2x –1) to the 5th power, (2x – y) to the 6th power, (x² + 1) to the 5th power]

B2. Demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems

Identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation
Determine the formula for the general term of an arithmetic sequence [i.e., t base n = a + (n –1)d] or geometric sequence (i.e., t base n = a(r to the (n–1) power)), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a sequence
Determine the formula for the sum of an arithmetic or geometric series, through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate the sum of a given number of consecutive terms
Solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications

B3. Make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities

Make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology (e.g., use a spreadsheet or graphing calculator to make simple interest calculations, determine first differences in the amounts over time, and graph amount versus time)
Make and describe connections between compound interest, geometric sequences, and exponential growth, through investigation with technology (e.g., use a spreadsheet to make compound interest calculations, determine finite differences in the amounts over time, and graph amount versus time)
Solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), the principal, P (also referred to as present value, PV), or the interest rate per compounding period, i, using the compound interest formula in the form A = P ((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]
Determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula in the form A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]; describe strategies (e.g., guessing and checking; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems
Explained he meaning of the term annuity, and determine the relationships between ordinary simple annuities (i.e., annuities in which payments are made at the end of each period, and compounding and payment periods are the same), geometric series, and exponential growth, through investigation with technology (e.g., use a spreadsheet to determine and graph the future value of an ordinary simple annuity for varying numbers of compounding periods; investigate how the contributions of each payment to the future value of an ordinary simple annuity are related to the terms of a geometric series)
Determine, through investigation using technology (e.g., the TVM Solver on a graphing calculator, online tools), the effects of changing the conditions (i.e., the payments, the frequency of the payments, the interest rate, the compounding period) of ordinary simple annuities (e.g., long-term savings plans, loans)
Solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan)

C. Trigonometric Functions

C1. Determine the values of the trigonometric ratios for angles less than 360°; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law

Determine the exact values of the sine, cosine, and tangent of the special angles: 0°, 30°, 45°, 60°, and 90°
Determine the values of the sine, cosine, and tangent of angles from 0° to 360°, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles)
Determine the measures of two angles from 0° to 360° for which the value of a given trigonometric ratio is the same
Define the secant, cosecant, and cotangent ratios for angles in a right triangle in terms of the sides of the triangle (e.g., sec A = hypotenuse/adjacent), and relate these ratios to the cosine, sine, and tangent ratios (e.g., sec A = 1/cos A)
Prove simple trigonometric identities, using the Pythagorean identity sin²x + cos²x = 1; the quotient identity tanx = sinx/cosx; and the reciprocal identities secx = 1/cosx, cscx = 1/sinx, and cotx = 1/tanx
Pose problems involving right triangles and oblique triangles in two-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)
Pose problems involving right triangles and oblique triangles in three-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law

C2. Demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions

Describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numerical or graphical representation
Predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements; predicting natural gas consumption in Ontario from previous consumption)
Make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0° to 360° and the corresponding sine ratios or cosine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function f(x) = sinx or f(x) = cosx, and explaining why the relationship is a function
Sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals)
Determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
Determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = a sin(k(x – d)) + c or f(x) = a cos(k(x – d)) + c
Sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) = sinx and f (x) = cosx, and state the domain and range of the transformed functions
Represent a sinusoidal function with an equation, given its graph or its properties

C3. Identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications

Collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data
Identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range
Determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles
Predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied (e.g., varying the conditions, such as speed and direction, when walking in a circle in front of a motion sensor)
Pose problems based on applications involving a sinusoidal function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation