MPM1D
Principles of Mathematics, Grade 9, Academic

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Course Description

This course enables students to develop an understanding of mathematical concepts related to algebra, analytic geometry, and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a linear relation. They will also explore relationships that emerge from the measurement of three-dimensional figures and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multi-step problems.

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Overall Provincial Curriculum Expectations

A: Number Sense and Algebra

1. demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
2. manipulate numerical and polynomial expressions, and solve first-degree equations.

B: Linear Relations

1. apply data-management techniques to investigate relationships between two variables;
2. demonstrate an understanding of the characteristics of a linear relation;
3. connect various representations of a linear relation.

C: Analytic Geometry

1. determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
2. determine, through investigation, the properties of the slope and y-intercept of a linear relation;
3. solve problems involving linear relations.

D: Measurement and Geometry

1. determine, through investigation, the optimal values of various measurements;
2. solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures;
3. verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.

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Specific Provincial Curriculum Expectations

A1: Operating with Exponents

• – substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate expressions involving natural-number exponents with rational-number bases
• – describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or $x^2$; volume, which is three dimensional, can be represented by (x)(x)(x), $(x^2)(x)$, or $x^3$];
• – derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents;
• – extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.

A2: Manipulating Expressions and Solving Equations

• – simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
• – solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion;
• – relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations;
• – add and subtract polynomials with up to two variables [e.g., (2x – 5) + (3x + 1), $(3x^2y + 2xy^2) + \\ (4x^2y – 6xy^2)$], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
• – multiply a polynomial by a monomial involving the same variable [e.g., 2x(x + 4), $2x^2(3x^2 – 2x + 1)$], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil);
• – expand and simplify polynomial expressions involving one variable [e.g., 2x(4x + 1) – 3x(x + 2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
• – solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
• – rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement)
• – solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods

B1: Using Data Management to Investigate Relationships

• – interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant [e.g., on a scatter plot of height versus age, interpret the point (13, 150) as representing a student who is 13 years old and 150 cm tall; identify points on the graph that represent students who are taller and younger than this student]
• – pose problems, identify variables, and formulate hypotheses associated with relationships between two variables
• – design and carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g.,making tables, drawing graphs)
• – describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses (e.g., describe the trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your hypothesis? Identify and explain any outlying pieces of data. Suggest a formula that relates the variables. How might you vary this experiment to examine other relationships?)

B2: Understanding Characteristics of Linear Relations

• – construct tables of values, graphs, and equations, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil), to represent linear relations derived from descriptions of realistic situations
• – construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools (e.g., spreadsheets, graphing software, graphing calculators, paper and pencil), for linearly related and non-linearly related data collected from a variety of sources (e.g., experiments, electronic secondary sources, patterning with concrete materials)
• – identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or non-linear;
• – compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or an equation)
• – determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., using a movable line in dynamic statistical software; using a process of trial and error on a graphing calculator; deterand error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot)

B3: Connecting Various Representations of Linear Relations

• – determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation
• – describe a situation that would explain the events illustrated by a given graph of a relationship between two variables
• – determine other representations of a linear relation, given one representation (e.g., given a numeric model, determine a graphical model and an algebraic model; given a graph, determine some points on the graph and determine an algebraic model);
• – describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the fixed costs are altered, and make the corresponding changes to the equation)

C1: Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph

• – determine, through investigation, the characteristics that distinguish the equation of a straight line from the equations of nonlinear relations (e.g., use a graphing calculator or graphing software to graph a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; connect an equation of degree one to a linear relation);
• – identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b;
• – express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.

C2: Investigating the Properties of Slope

• – determine, through investigation, various formulas for the slope of a line segment or a line, and use the formulas to determine the slope of a line segment or a line;
• – identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;
• – determine, through investigation, connections among the representations of a constant rate of change of a linear relation
• – identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate

C3: Using the Properties of Linear Relations to Solve Problems

• – graph lines by hand, using a variety of techniques (e.g., graph $y =\frac{2}{3} x – 4$ using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);
• – determine the equation of a line from information about the line (e.g., the slope and y-intercept; the slope and a point; two points)
• – describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation
• – identify and explain any restrictions on the variables in a linear relation arising from a realistic situation (e.g., in the relation C = 50 + 25n,C is the cost of holding a party in a hall and n is the number of guests; n is restricted to whole numbers of 100 or less, because of the size of the hall, and C is consequently restricted to $50 to $2550);
• – determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application

D1: Investigating the Optimal Values of Measurements

• – determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, toothpicks, a pre-made dynamic geometry sketch), and by examining various values of the area as the side lengths change and the perimeter remains constant;
• – determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, a premade dynamic geometry sketch), and by examining various values of the side lengths and the perimeter as the area stays constant;
• – identify, through investigation with a variety of tools (e.g. concrete materials, computer software), the effect of varying the dimensions on the surface area [or volume] of square-based prisms and cylinders, given a fixed volume [or surface area];
• – explain the significance of optimal area, surface area, or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);
• – pose and solve problems involving maximization and minimization of measurements of geometric shapes and figures (e.g., determine the dimensions of the rectangular field with the maximum area that can be enclosed by a fixed amount of fencing, if the fencing is required on only three sides)

D2: Solving Problems Involving Perimeter, Area, Surface Area, and Volume

• – relate the geometric representation of the Pythagorean theorem and the algebraic representation $a^2 + b^2 = c^2$;
• – solve problems using the Pythagorean theorem, as required in applications (e.g., calculate the height of a cone, given the radius and the slant height, in order to determine the volume of the cone);
• – solve problems involving the areas and perimeters of composite two-dimensional shapes (i.e., combinations of rectangles, triangles, parallelograms, trapezoids, and circles)
• – develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere
• – determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a squarebased pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);
• – solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures

D3: Investigating and Applying Geometric Relationships

• – determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials), and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons
• – determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the figure that results from joining the midpoints of the sides of a quadrilateral is a parallelogram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizontal support beam that is attached half way up the sloping sides);
• – pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms (e.g., written explanations, diagrams, dynamic sketches, formulas, tables)
• – illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example, with or without the use of dynamic geometry software

MPM2D
Principles of Mathematics, Grade 10, Academic

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Course Description

This course enables students to broaden their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relations and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. Students will reason mathematically and communicate their thinking as they solve multi-step problems.

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Overall Provincial Curriculum Expectations

A: Quadratic Relations of the Form $y=ax^2+bx+c$

1. determine the basic properties of quadratic relations;
2. relate transformations of the graph of $y = x^2$ to the algebraic representation $y = a(x - h)^2 + k$;
3. solve quadratic equations and interpret the solutions with respect to the corresponding relations;

B: Analytic Geometry

1. model and solve problems involving the intersection of two straight lines;
2. solve problems using analytic geometry involving properties of lines and line segments;
3. verify geometric properties of triangles and quadrilaterals, using analytic geometry.

C: Trigonometry

1. use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity;
2. solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
3. solve problems involving acute triangles, using the sine law and the cosine law.

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Specific Provincial Curriculum Expectations

A1: Investigating the Basic Properties of Quadratic Relations

• – collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit, if appropriate, with or without the use of technology
• – determine, through investigation with and without the use of technology, that a quadratic relation of the form $y = ax^2 + bx + c, a\neq 0$ can be graphically represented as a parabola, and that the table of values yields a constant second difference
• – identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum value), and use the appropriate terminology to describe them;
• – compare, through investigation using technology, the features of the graph of $y = x^2$ and the graph of $y = 2^x$, and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for $y = 2^x$; by applying the exponent rules for multiplication and division)

A2: Relating the Graph of $y=x^2$ And Its Transformations

• – identify, through investigation using technology, the effect on the graph of $y = x^2$ of transformations (i.e., translations, reflections in the x-axis, vertical stretches or compressions) by considering separately each parameter $a, h, k$ [i.e., investigate the effect on the graph of $y = x^2$ of $a, h, k$ in $y = x^2 + k, y = (x – h)^2$ and $y = ax^2$];
• – explain the roles of $a, h, k$ in $y = a(x – h )^2 + k$, using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of symmetry;
• – sketch, by hand, the graph of $y = a(x – h )^2 + k$ by applying transformations to the graph of $y = x^2$
• – determine the equation, in the form $y = a(x – h)^2 + k$, of a given graph of a parabola

A3: Solving Quadratic Equations

• – expand and simplify second-degree polynomial expressions [e.g., $(2x + 5)^2$, (2x – y)(x + 3y)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);
• – factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., $2x^2 + 4x$, $2x – 2y + ax – ay$, $x2 – x – 6$, $2a^2 + 11a + 5$, $4x^2 – 25$], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning);
• – determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts (i.e., the zeros) of the graph of the corresponding quadratic relation, expressed in the form y = a(x – r)(x – s);
• – interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corresponding relations;
• – express $y = ax^2 + bx + c$ in the form $y = a(x – h)^2 + k$ by completing the square in situations involving no fractions, using a variety of tools (e.g. concrete materials, diagrams, paper and pencil);
• – sketch or graph a quadratic relation whose equation is given in the form $y = ax^2 + bx + c$, using a variety of methods (e.g., sketching $y = x^2 – 2x – 8$ using intercepts and symmetry; sketching $y = 3x^2 – 12x + 1$ by completing the square and applying transformations; graphing $h = –4.9t^2 + 50t + 1.5$ using technology);
• – explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]);
• – solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing)

A4: Solving Problems Involving Quadratic Equations

• – determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calculators or graphing software) or from its defining equation (i.e., by applying algebraic techniques);
• – solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology (e.g., given the graph or the equation of a quadratic relation representing the height of a ball over elapsed time, answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3 m?)

B1: Using Linear Systems to Solve Problems

• – solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination
• – solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method

B2: Solving Problems Involving Properties of Line Segments

• – develop the formula for the midpoint of a line segment, and use this formula to solve problems (e.g., determine the coordinates of the midpoints of the sides of a triangle, given the coordinates of the vertices, and verify concretely or by using dynamic geometry software);
• – develop the formula for the length of a line segment, and use this formula to solve problems (e.g., determine the lengths of the line segments joining the midpoints of the sides of a triangle, given the coordinates of the vertices of the triangle, and verify using dynamic geometry software);
• – develop the equation for a circle with centre (0, 0) and radius r, by applying the formula for the length of a line segment;
• – determine the radius of a circle with centre (0, 0), given its equation; write the equation of a circle with centre (0, 0), given the radius; and sketch the circle, given the equation in the form $x^2 + y^2 = r^2$;
• – solve problems involving the slope, length, and midpoint of a line segment (e.g., determine the equation of the right bisector of a line segment, given the coordinates of the endpoints; determine the distance from a given point to a line whose equation is given, and verify using dynamic geometry software)

B3: Using Analytic Geometry to Verify Geometric Properties

• – determine, through investigation (e.g., using dynamic geometry software, by paper folding), some characteristics and properties of geometric figures (e.g., medians in a triangle, similar figures constructed on the sides of a right triangle);
• – verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures (e.g., verify that two lines are perpendicular, given the coordinates of two points on each line; verify, by determining side length, that a triangle is equilateral, given the coordinates of the vertices);
• – plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property (e.g., given the coordinates of the vertices of a triangle, verify that the line segment joining the midpoints of two sides of the triangle is parallel to the third side and half its length, and check using dynamic geometry software; given the coordinates of the vertices of a rectangle, verify that the diagonals of the rectangle bisect each other)

C1: Investigating Similarity and Solving Problems Involving Similar Triangles

• – verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);
• – describe and compare the concepts of similarity and congruence;
• – solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying)

C2: Solving Problems Involving the Trigonometry of Right Triangles

• – determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios
• – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
• – solve problems involving the measures of sides and angles in right triangles in reallife applications (e.g., in surveying, in navigating, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem

C3: Solving Problems Involving the Trigonometry of Acute Triangles

• – explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law and identify the application of solving systems of equations [student reproduction of the development of the formula is not required]);
• – explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the
• cosine ratio [student reproduction of the development of the formula is not required]);
• – determine the measures of sides and angles in acute triangles, using the sine law and the cosine law
• – solve problems involving the measures of sides and angles in acute triangles.

MCR3U
Functions, Grade 11, University Preparation

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Course Description

This course introduces the mathematical concept of the function by extending students' experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.

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Overall Provincial Curriculum Expectations

A: Characteristics of Functions

1. demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;
2. 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;
3. demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.

B: Exponential Functions

1. evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;
2. make connections between the numeric, graphical, and algebraic representations of exponential functions;
3. identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.

C: Discrete Functions

1. demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal’s triangle;
2. demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems;
3. make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.

D: Trigonometric Functions

1. determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;
2. demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;
3. identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.

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Specific Provincial Curriculum Expectations

A1: Representing Functions

• 1.1 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 verticalline test)
• 1.2 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 $f(\frac{1}{2})$, given $f(x) = 2x^2 + 3x – 1$]
• 1.3 explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions $f(x)=x,~x^2,~\sqrt{x},~ \frac{1}{x}$; describe the domain and range of a function appropriately (e.g., for $y=x^2+1$, the domain is the set of all real numbers and the range is the set $y\geq 1$); and explain any restrictions on the domain and range in contexts arising from real-world applications
• 1.4 relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations)
• 1.5 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 $y = x$)
• 1.6 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
• 1.7 determine, using function notation when appropriate, the algebraic representation of the inverse of a linear or quadratic function, given the algebraic representation of the function 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)
• 1.8 determine, through investigation using technology, the roles of the parameters $a, k, d, c$ in functions of the form $y = af (k(x – d)) + c$, and describe these roles in terms of transformations on the graphs of $f(x)=x,~x^2,~\sqrt{x},~ \frac{1}{x}$ (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
• 1.9 sketch graphs of $y = af (k(x – d)) + c$ by applying one or more transformations to the graphs of $f(x)=x,~x^2,~\sqrt{x},~ \frac{1}{x}$, and state the domain and range of the transformed functions

A2: Solving Problems Involving Quadratic Functions

• 2.1 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)
• 2.2 determine the maximum or minimum value of a quadratic function whose equation is given in the form $f(x) = ax^2 + bx + c$, using an algebraic method (e.g., completing the square; factoring to determine the zeros and averaging the zeros)
• 2.3 solve problems involving quadratic functions arising from real-world applications and represented using function notation
• 2.4 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
• 2.5 solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determine 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: Determining Equivalent Algebraic Expressions

• 3.1 simplify polynomial expressions by adding, subtracting, and multiplying
• 3.2 verify, through investigation with and without technology, that $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$, $a,b\geq 0$, and use this relationship to simplify radicals and radical expressions obtained by adding, subtracting, and multiplying
• 3.3 simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values
• 3.4 determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values).

B1: Representing Exponential Functions

• 1.1 graph, with and without technology, an exponential relation, given its equation in the form $y = a^x$ ($a > 0, ~a\neq 1$), define this relation as the function $f(x) = a^x$, and explain why it is a function
• 1.2 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. $x^{m/n}$ where $x>0$ and $m$ and $n$ are integers)
• 1.3 simplify algebraic expressions containing integer and rational exponents, and evaluate numeric expressions containing integer and rational exponents and rational bases
• 1.4 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 $f(x) = a^x$, function machines]

B2: Connecting Graphs and Equations of Exponential Functions

• 2.1 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)
• 2.2 determine, through investigation using technology, the roles of the parameters $a, k, d, c$ in functions of the form $y = af (k(x – d)) + c$, and describe these roles in terms of transformations on the graph of $f(x) = a^x$ (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
• 2.3 sketch graphs of $y = af (k(x – d)) + c$ by applying one or more transformations to the graph of $f(x) = a^x$, and state the domain and range of the transformed functions
• 2.4 determine, through investigation using technology, that the equation of a given exponential function can be expressed using different bases [e.g., $f(x) = 9^x$ can be expressed as $f(x) =3^{2x}$ ], and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws)
• 2.5 represent an exponential function with an equation, given its graph or its properties

B3: Solving Problems Involving Exponential Functions

• 3.1 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, E-STAT), and graph the data
• 3.2 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)
• 3.3 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

C1: Representing Sequences

• 1.1 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]
• 1.2 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, $\cdots$), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)
• 1.3 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
• 1.4 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,$\cdots$ as $t_1 = 2; t_n = 2t_{n – 1}$, as $f(n)=2^n$ or as $t_n=2^n$ 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)
• 1.5 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)
• 1.6 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

C2: Investigating Arithmetic and Geometric Sequences and Series

• 2.1 identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation
• 2.2 determine the formula for the general term of an arithmetic sequence [i.e., $t_n = a + (n –1)d $] or geometric sequence (i.e., $t_n = ar^{n – 1}$), 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
• 2.3 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
• 2.4 solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications

C3: Solving Problems Involving Financial Applications

• 3.1 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)
• 3.2 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)
• 3.3 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 $A = P(1 + i)^n$ [or $FV = PV(1 + i)^n$ ]
• 3.4 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 $A = P(1 + i)^n$ [or $FV = PV(1 + i)^n$ ]; 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
• 3.5 explain the 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)
• 3.6 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)
• 3.7 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)

D1: Determining and Applying Trigonometric Ratios

• 1.1 determine the exact values of the sine, cosine, and tangent of the special angles: 0º, 30º, 45º, 60º, and 90º
• 1.2 determine the values of the sine, cosine, and tangent of angles from 0º to 360º, 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)
• 1.3 determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the same
• 1.4 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 =\frac{hypotenuse}{adjacent} $), and relate these ratios to the cosine, sine, and tangent ratios (e.g., $\sec A =\frac{1}{\cos A} $)
• 1.5 prove simple trigonometric identities, using the Pythagorean identity $\sin^2 x+\cos^2 x=1$; the quotient identity $\tan x=\frac{\sin x}{\cos x}$; and the reciprocal identities
• 1.6 pose problems involving right triangles and oblique triangles in twodimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)
• 1.7 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

D2: Connecting Graphs of Equations and Sinusoidal Functions

• 2.1 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 numeric or graphical representation
• 2.2 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)
• 2.3 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 0º to 360º 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 $f(x) =\sin x$ or $f(x) =\cos x$, and explaining why the relationship is a function
• 2.4 sketch the graphs of $f(x) =\sin x$ and $f(x) =\cos x$ 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)
• 2.5 determine, through investigation using technology, the roles of the parameters $a, k, d, c$ in functions of the form $y =af (k(x – d)) + c$, where $f(x) =\sin x$ or $f(x) =\cos x$ with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of $f(x) =\sin x$ and $f(x) =\cos x$ (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
• 2.6 determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form $y = a\sin(k(x – d)) + c$ or $y = a\cos(k(x – d)) + c$
• 2.7 sketch graphs of $y = af (k(x – d)) + c$ by applying one or more transformations to the graphs of $f(x) =\sin x$ and $f(x) =\cos x$, and state the domain and range of the transformed functions
• 2.8 represent a sinusoidal function with an equation, given its graph or its properties

D3: Solving Problems Involving Trigonometric Functions

• 3.1 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
• 3.2 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
• 3.3 determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles
• 3.4 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)
• 3.5 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

MHF4U
Advanced Functions, Grade 12, University Preparation

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Course Description

This course extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.

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Overall Provincial Curriculum Expectations

A: Exponential and Logarithmic Functions

1. demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
2. identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
3. solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

B: Trigonometric Functions

1. demonstrate an understanding of the meaning and application of radian measure;
2. make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
3. solve problems involving trigonometric equations and prove trigonometric identities.

C: Polynomial and Rational Functions

1. identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
2. identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
3. solve problems involving polynomial and simple rational equations graphically and algebraically;
4. demonstrate an understanding of solving polynomial and simple rational inequalities.

D: Characteristics of Functions

1. demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
2. determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
3. compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.

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Specific Provincial Curriculum Expectations

A1: Evaluating Logarithms

• 1.1 recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions
• 1.2 determine, with technology, the approximate logarithm of a number to any base, including base 10 (e.g., by reasoning that $\log_{3} 29$ is between 3 and 4 and using systematic trial to determine that $\log_{3} 29$ is approximately 3.07)
• 1.3 make connections between related logarithmic and exponential equations, and solve simple exponential equations by rewriting them in logarithmic form
• 1.4 make connections between the laws of exponents and the laws of logarithms, verify the laws of logarithms with or without technology (e.g., use patterning to verify the quotient law for logarithms by evaluating expressions such as $\log_{10} 1000 – \log_{10} 100$ and then rewriting the answer as a logarithmic term to the same base), and use the laws of logarithms to simplify and evaluate numerical expressions

A2: Connecting Graphs and Equations of Logarithmic Functions

• 2.1 determine, through investigation with technology (e.g., graphing calculator, spreadsheet) and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, increasing/decreasing behaviour) of the graphs of logarithmic functions of the form $f(x) = \log_{b} x$, and make connections between the algebraic and graphical representations of these logarithmic functions
• 2.2 recognize the relationship between an exponential function and the corresponding logarithmic function to be that of a function and its inverse, deduce that the graph of a logarithmic function is the reflection of the graph of the corresponding exponential function in the line $y = x$, and verify the deduction using technology
• 2.3 determine, through investigation using technology, the roles of the parameters $d$ and $c$ in functions of the form $y = \log_{10} (x – d) + c$ and the roles of the parameters $a$ and $k$ in functions of the form $y = a\log_{10} (kx)$, and describe these roles in terms of transformations on the graph of $f(x) = \log_{10}x$ (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
• 2.4 pose problems based on real-world applications of exponential and logarithmic functions (e.g., exponential growth and decay, the Richter scale, the pH scale, the decibel scale), 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

A3: Solving Exponential and Logarithmic Equations

• 3.1 recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
• 3.2 solve exponential equations in one variable by determining a common base (e.g., solve $4^x = 8^{x+3}$ by expressing each side as a power of 2) and by using logarithms (e.g., solve $4^x = 8^{x+3}$ by taking the logarithm base 2 of both sides), recognizing that logarithms base 10 are commonly used (e.g., solving $3^x = 7$ by taking the logarithm base 10 of both sides)
• 3.3 solve simple logarithmic equations in one variable algebraically
• 3.4 solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications

B1: Understanding and Applying Radian Measure

• 1.1 recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure
• 1.2 represent radian measure in terms of $\pi$ (e.g., $\frac{\pi}{3} $ radians, $2\pi$ radians) and as a rational number (e.g., 1.05 radians, 6.28 radians)
• 1.3 determine, with technology, the primary trigonometric ratios (i.e., sine, cosine, tangent) and the reciprocal trigonometric ratios (i.e., cosecant, secant, cotangent) of angles expressed in radian measure
• 1.4 determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles $0, \frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}$, and their multiples less than or equal to $2\pi$

B2: Connecting Graphs and Equations of Trigonometric Functions

• 2.1 sketch the graphs of $f(x) = \sin x$ and $f(x) = \cos x$ for angle measures expressed in radians, and determine and describe some key properties (e.g., period of $2\pi$, amplitude of 1) in terms of radians
• 2.2 make connections between the tangent ratio and the tangent function by using technology to graph the relationship between angles in radians and their tangent ratios and defining this relationship as the function $f(x) = \tan x$, and describe key properties of the tangent function
• 2.3 graph, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions (i.e., cosecant, secant, cotangent) for angle measures expressed in radians, determine and describe key properties of the reciprocal functions (e.g., state the domain, range, and period, and identify and explain the occurrence of asymptotes), and recognize notations used to represent the reciprocal functions
• 2.4 determine the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form $y = a \sin (k(x – d)) + c$ or $y = a \cos(k(x – d)) + c$, with angles expressed in radians
• 2.5 sketch graphs of $y = a \sin (k(x – d)) + c$ and $y = a \cos(k(x – d)) + c$ by applying transformations to the graphs of $f(x) = \sin x$ and $f(x) = \cos x$ with angles expressed in radians, and state the period, amplitude, and phase shift of the transformed functions
• 2.6 represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
• 2.7 pose problems based on applications involving a trigonometric function with domain expressed in radians (e.g., seasonal changes in temperature, heights of tides, hours of daylight, displacements for oscillating springs), and solve these and other such problems by using a given graph or a graph generated with or without technology from a table of values or from its equation

B3: Solving Trigonometric Equations

• 3.1 recognize equivalent trigonometric expressions [e.g., by using the angles in a right triangle to recognize that $\sin x$ and $\cos (\frac{\pi}{2} – x)$ are equivalent; by using transformations to recognize that $\cos (x + \frac{\pi}{2})$ and $–\sin x$ are equivalent], and verify equivalence using graphing technology
• 3.2 explore the algebraic development of the compound angle formulas (e.g., verify the formulas in numerical examples, using technology; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]), and use the formulas to determine exact values of trigonometric ratios
• 3.3 recognize that trigonometric identities are equations that are true for every value in the domain (i.e., a counter-example can be used to show that an equation is not an identity), prove trigonometric identities through the application of reasoning skills, using a variety of relationships, and verify identities using technology
• 3.4 solve linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from $0$ to $2\pi$, and solve related problems

C1: Connecting Graphs and Equations of Polynomial Functions

• 1.1 recognize a polynomial expression (i.e., a series of terms where each term is the product of a constant and a power of x with a nonnegative integral exponent, such as $x^3 – 5x^2 + 2x – 1$); recognize the equation of a polynomial function, give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions
• 1.2 compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., compare finite differences in tables of values; investigate the effect of the degree of a polynomial function on the shape of its graph and the maximum number of x-intercepts; investigate the effect of varying the sign of the leading coefficient on the end behaviour of the function for very large positive or negative x-values)
• 1.3 describe key features of the graphs of polynomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or negative x-values)
• 1.4 distinguish polynomial functions from sinusoidal and exponential functions [e.g., $f(x) = \sin x$, $g(x) = 2^x$ ], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions
• 1.5 make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g., f(x) = 2(x – 3)(x + 2)(x – 1)] and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features (e.g., by determining intercepts and end behaviour; by locating positive and negative regions using test values between and on either side of the x-intercepts)
• 1.6 determine, through investigation using technology, the roles of the parameters $a, k, d, c$ in functions of the form $y = af (k(x – d)) + c$, and describe these roles in terms of transformations on the graphs of $f(x) = x^3$ and $f(x) = x^4$ (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
• 1.7 determine an equation of a polynomial function that satisfies a given set of conditions (e.g., degree of the polynomial, intercepts, points on the function), using methods appropriate to the situation (e.g., using the x-intercepts of the function; using a trial-and-error process with a graphing calculator or graphing software; using finite differences), and recognize that there may be more than one polynomial function that can satisfy a given set of conditions (e.g., an infinite number of polynomial functions satisfy the condition that they have three given x-intercepts)
• 1.8 determine the equation of the family of polynomial functions with a given set of zeros and of the member of the family that passes through another given point [e.g., a family of polynomial functions of degree 3 with zeros 5, –3, and –2 is defined by the equation f(x) = k(x – 5)(x + 3)(x + 2), where $k$ is a real number, $k \neq 0$; the member of the family that passes through (–1, 24) is f(x) = –2(x – 5)(x + 3)(x + 2)]
• 1.9 determine, through investigation, and compare the properties of even and odd polynomial functions [e.g., symmetry about the y-axis or the origin; the power of each term; the number of x-intercepts; $f(x) = f(– x)$ or $f(– x) = – f (x)$], and determine whether a given polynomial function is even, odd, or neither

C2: Connecting Graphs and Equations of Rational Functions

• 2.1 determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions
• 2.2 determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that have linear expressions in the numerator and denominator, and make connections between the algebraic and graphical representations of these rational functions
• 2.3 sketch the graph of a simple rational function using its key features, given the algebraic representation of the function

C3: Solving Polynomial and Rational Equations

• 3.1 make connections, through investigation using technology (e.g., computer algebra systems), between the polynomial function f(x), the divisor x – a, the remainder from the division $\frac{f(x)}{x-a}$, and f(a) to verify the remainder theorem and the factor theorem
• 3.2 factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem)
• 3.3 determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this connection
• 3.4 solve polynomial equations in one variable, of degree no higher than four (e.g., $2x^3 – 3x^2 + 8x – 12 = 0$), by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem), and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding polynomial function)
• 3.5 determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection
• 3.6 solve simple rational equations in one variable algebraically, and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding rational function)
• 3.7 solve problems involving applications of polynomial and simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem, such as determining the values of k for which the function $y= x^3 + 6x^2 + kx – 4$ gives the same remainder when divided by x – 1 and x + 2]

C4: Solving Inequalities

• 4.1 explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one variable, and demonstrate that given solutions satisfy an inequality
• 4.2 determine solutions to polynomial inequalities in one variable [e.g., solve $y\geq 0$, where $y = x^3 – x^2 + 3x – 9$] and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities
• 4.3 solve linear inequalities and factorable polynomial inequalities in one variable (e.g., $x^3 + x^2 > 0$) in a variety of ways (e.g., by determining intervals using x-intercepts and evaluating the corresponding function for a single x-value within each interval; by factoring the polynomial and identifying the conditions for which the product satisfies the inequality), and represent the solutions on a number line or algebraically

D1: Understanding Rates of Change

• 1.1 gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change (e.g., in words, numerically, graphically, algebraically)
• 1.2 recognize that the rate of change for a function is a comparison of changes in the dependent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or changing by examining applications, including those arising from real-world situations (e.g., rate of change of the area of a circle as the radius increases, inflation rates, the rising trend in graduation rates among Aboriginal youth, speed of a cruising aircraft, speed of a cyclist climbing a hill, infection rates)
• 1.3 sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology (e.g., motion sensor) when possible
• 1.4 calculate and interpret average rates of change of functions (e.g., linear, quadratic, exponential, sinusoidal) arising from real-world applications (e.g., in the natural, physical, and social sciences), given various representations of the functions (e.g., tables of values, graphs, equations)
• 1.5 recognize examples of instantaneous rates of change arising from real-world situations, and make connections between instantaneous rates of change and average rates of change (e.g., an average rate of change can be used to approximate an instantaneous rate of change)
• 1.6 determine, through investigation using various representations of relationships (e.g., tables of values, graphs, equations), approximate instantaneous rates of change arising from real-world applications (e.g., in the natural, physical, and social sciences) by using average rates of change and reducing the interval over which the average rate of change is determined
• 1.7 make connections, through investigation, between the slope of a secant on the graph of a function (e.g., quadratic, exponential, sinusoidal) and the average rate of change of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point
• 1.8 determine, through investigation using a variety of tools and strategies (e.g., using a table of values to calculate slopes of secants or graphing secants and measuring their slopes with technology), the approximate slope of the tangent to a given point on the graph of a function (e.g., quadratic, exponential, sinusoidal) by using the slopes of secants through the given point (e.g., investigating the slopes of secants that approach the tangent at that point more and more closely), and make connections to average and instantaneous rates of change
• 1.9 solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods (e.g., by using graphing technology to graph a tangent and measure its slope)

D2: Combining Functions

• 2.1 determine, through investigation using graphing technology, key features (e.g., domain, range, maximum/minimum points, number of zeros) of the graphs of functions created by adding, subtracting, multiplying, or dividing functions
• 2.2 recognize real-world applications of combinations of functions (e.g., the motion of a damped pendulum can be represented by a function that is the product of a trigonometric function and an exponential function; the frequencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically
• 2.3 determine, through investigation, and explain some properties (i.e., odd, even, or neither; increasing/decreasing behaviours) of functions formed by adding, subtracting, multiplying, and dividing general functions
• 2.4 determine the composition of two functions [i.e., f(g(x))] numerically (i.e., by using a table of values) and graphically, with technology, for functions represented in a variety of ways (e.g., function machines, graphs, equations), and interpret the composition of two functions in real-world applications
• 2.5 determine algebraically the composition of two functions [i.e., f(g(x))], verify that f(g(x)) is not always equal to g( f(x)) [e.g., by determining f(g(x)) and g( f(x)), given f(x) = x + 1 and g(x) = 2x], and state the domain [i.e., by defining f(g(x)) for those x-values for which g(x) is defined and for which it is included in the domain of f(x)] and the range of the composition of two functions
• 2.6 solve problems involving the composition of two functions, including problems arising from real-world applications
• 2.7 demonstrate, by giving examples for functions represented in a variety of ways (e.g., function machines, graphs, equations), the property that the composition of a function and its inverse function maps a number onto itself
• 2.8 make connections, through investigation using technology, between transformations (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) of simple functions f(x) [e.g., $f(x) = x^3 + 20$, $f(x) = \sin x$, $f(x) = \log x$] and the composition of these functions with a linear function of the form $g(x) = A(x + B)$

D3: Using Function Models to Solve Problems

• 3.1 compare, through investigation using a variety of tools and strategies (e.g., graphing with technology; comparing algebraic representations; comparing finite differences in tables of values) the characteristics (e.g., key features of the graphs, forms of the equations) of various functions (i.e., polynomial, rational, trigonometric, exponential, logarithmic)
• 3.2 solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques
• 3.3 solve problems, using a variety of tools and strategies, including problems arising from real-world applications, by reasoning with functions and by applying concepts and procedures involving functions (e.g., by constructing a function model from data, using the model to determine mathematical results, and interpreting and communicating the results within the context of the problem).