A.Solving quadratic equations:
Students will

1.pose problems involving quadratic relations arising from real-world applications and represented by tables of values and graphs, and solve these and other such problems (e.g., “From the graph of the height of a ball versus time, can you tell me how high the ball was thrown and the time when it hit the ground?”).

2.represent situations (e.g., the area of a picture frame of variable width) using quadratic expressions in one variable and expand and simplify quadratic expression in one variable [e.g., .

3.factor quadratic expressions in one variable, including those for which a ≠ 1 (e.g., ), differences of squares (e.g., ), and perfect square trinomials (e.g.,  ), by selecting and applying an appropriate strategy.

4.solve quadratic equations by selecting and applying a factoring strategy.

5.determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the graph of the corresponding quadratic relation.

6.explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numeric example; follow a demonstration of the algebraic development, with technology, such as computer algebra systems, or without technology [student reproduction of the development of the general case is not required]), and apply the formula to solve quadratic equations, using technology.

7.relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if ).

8.relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if ).

9.determine the real roots of a variety of quadratic equations (e.g., 100x² = 115x + 35) and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula).

B.Connecting graphs and equations of quadratic functions:
Students will

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., using the vertical-line test).

2.substitute into and evaluate linear and quadratic functions represented using function notation [e.g., evaluate , given ], including functions arising from real-world applications.

3.explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of linear and quadratic functions, and describe the domain and range of a function appropriately (e.g., for , the domain is the set of real numbers, and the range is .

4.explain any restrictions on the domain and the range of a quadratic function in contexts arising from real-world applications.

5.determine, through investigation using technology, the roles of a, h, and k in quadratic functions of the form , and describe these roles in terms of transformations on the graph of (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

6.sketch graphs of by applying one or more transformations to the graph of .

7.express the equation of a quadratic function in the standard form , given the vertex form , and verify, using graphing technology, that these forms are equivalent representations.

8.express the equation of a quadratic function in the vertex form , given the standard form , by completing the square (e.g., using algebra tiles or diagrams; algebraically), including cases where is a simple rational number (e.g., , ) and verify, using graphing technology, that these forms are equivalent representations.

9.sketch graphs of quadratic functions in the factored form by using the x-intercepts to determine the vertex.

10.describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form , the vertex form , and the factored form of a quadratic function.

11.sketch the graph of a quadratic function whose equation is given in the standard form by using a suitable strategy (e.g., completing the square and finding the vertex; factoring, if possible, to locate the x-intercepts), and identify the key features of the graph (e.g., the vertex, the x– and y-intercepts, the equation of the axis of symmetry, the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing).

C.Solving problems involving quadratic functions:
Students will

1.collect data that can be modelled as a quadratic function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials; measurement tools such as measuring tapes, electronic probes, motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data.

2.determine, through investigation using a variety of strategies (e.g., applying properties of quadratic functions such as the x-intercepts and the vertex; using transformations), the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology (e.g., graphing software, graphing calculator).

3.solve problems arising from real-world applications, given the algebraic representation of a quadratic function (e.g., given the equation of a quadratic function representing the height of a ball over elapsed time, answer questions that involve the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement).