Page 61 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. QUADRATIC FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1. expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;
2. demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;
3. solve problems involving quadratic functions, including problems arising from real-world applications.
SPECIFIC EXPECTATIONS
1. Solving Quadratic Equations
By the end of this course, students will:
1.1 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?”)
1.2 represent situations (e.g., the area of a picture frame of variable width) using quadratic expressions in one variable, and expand
and simplify quadratic expressions in one variable [e.g., 2x(x + 4) – (x + 3)2]*
1.3 factor quadratic expressions in one variable, including those for which a ≠ 1 (e.g.,
3x2 + 13x – 10), differences of squares
(e.g., 4x2 – 25), and perfect square trinomials (e.g., 9x2 + 24x + 16), by selecting and applying an appropriate strategy*
Sample problem: Factor 2x2 – 12x + 10.
1.4 solve quadratic equations by selecting and
applying a factoring strategy
1.5 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
Sample problem: The profit, P, of a video company, in thousands of dollars, is given byP=–5x2 +550x–5000,wherexisthe amount spent on advertising, in thousands of dollars. Determine, by factoring and by graphing, the amount spent on advertising that will result in a profit of $0. Describe the connection between the two strategies.
1.6 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
1.7 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 b2 – 4ac < 0)
1.8 determine the real roots of a variety of quad- ratic equations (e.g., 100x2 = 115x + 35), and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula)
Sample problem: Generate 10 quadratic equa- tions by randomly selecting integer values fora,b,andcinax2 +bx+c=0.Solvethe
QUADRATIC FUNCTIONS
*The knowledge and skills described in this expectation may initially require the use of a variety of learning tools (e.g., computer algebra systems, algebra tiles, grid paper).
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Functions and Applications
MCF3M