Page 48 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

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2. Solving Problems Involving Quadratic Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
and range of the inverse relation, and investi- gate connections to the domain and range of
the functions g(x) = √x and h(x) = –√x.
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 [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)
Sample problem: Given the equations of several linear functions, graph the functions and their inverses, determine the equations of the inverses, and look for patterns that connect the equation of each linear function with the equation of the inverse.
1.8 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) = x2, f(x) = √x, and
f(x) = 1 (i.e., translations; reflections in the x
axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
Sample problem: Investigate the graph
f(x) = 3(x – d)2 + 5 for various values of d, using technology, and describe the effects of changing d in terms of a transformation.
1.9 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) = √x, and f (x) = 1x , and state the domain and range of the transformed functions
Sample problem: Transform the graph of f(x) to sketch g(x), and state the domain and range of each function, for the following:
f(x) = √x, g(x) = √x – 4; f(x) = 1 , x g(x)=– 1 .
x+1
By the end of this course, students will:
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)
Sample problem: Investigate, using graphing technology and algebraic techniques, the transformations that affect the number of zeros for a given quadratic function.
2.2 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)
Sample problem: Explain how partially factoring f(x) = 3x2 – 6x + 5 into the form f(x) = 3x(x – 2) + 5 helps you determine the minimum of the function.
2.3 solve problems involving quadratic functions arising from real-world applications and represented using function notation
Sample problem: The profit, P(x), of a video company, in thousands of dollars, is given by P(x) = – 5x2 + 550x – 5000, where x is the amount spent on advertising, in thousands of dollars. Determine the maximum profit that the company can make, and the amounts spent on advertising that will result in a profit and that will result in a profit of at least $4 000 000.
2.4 determine, through investigation, the trans- formational 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
Sample problem: Determine the equation of the quadratic function that passes through (2, 5) if the roots of the corresponding
quadratic equation are 1 + √5 and 1 – √5.
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