Page 49 - Math 9-10 (2009)

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THE ONTARIO CURRICULUM, GRADES 9 AND 10: MATHEMATICS
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sketch, by hand, the graph of
y = a(x – h )2 + k by applying transforma- tions to the graph of y = x2 [Sample problem: Sketch the graph of
y = – (x – 3)2 + 4, and verify
using technology.];
determine the equation, in the form y=a(x–h)2 +k,ofagivengraphofa parabola.
y=ax2 +bx+c,usingavarietyof methods (e.g., sketching y = x2 – 2x – 8 using intercepts and symmetry; sketching y=3x2 –12x+1bycompletingthe square and applying transformations; graphing h = –4.9t2 + 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 demonstra- tion 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) (Sample problem: Solve
x2 + 10x + 16 = 0 by factoring,and verify algebraically. Solve x2 + x – 4 = 0 using the quadratic formula, and verify graphically using technology. Solve
–4.9t2 + 50t + 1.5 = 0 by graphing
h = –4.9t2 + 50t + 1.5 using technology.).
Solving Problems Involving
Quadratic Relations
By the end of this course, students will:
– determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calcu- lators or graphing software) or from its defining equation (i.e., by applying alge- braic techniques);
– solve problems arising from a realistic situ- ation represented by a graph or an equa- tion 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?).
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Solving Quadratic Equations
By the end of this course, students will:
– expand and simplify second-degree poly- nomial 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 differ- ences of squares [e.g., 2x2 + 4x, 2x–2y+ax–ay,x2 –x–6,
2a2 + 11a + 5, 4x2 – 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 qua- dratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corre- sponding relations;
– expressy=ax2 +bx+cintheform y=a(x–h)2 +kbycompletingthe 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