Page 62 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

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2. Connecting Graphs and Equations of Quadratic Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
equations using the quadratic formula. How many of the equations could you solve by factoring?
By the end of this course, students will:
2.1 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, equa- tions) and strategies (e.g., using the vertical- line test)
Sample problem: Investigate, using numeric and graphical representations, whether the relation x = y2 is a function, and justify your reasoning.
2.2 substitute into and evaluate linear and quadratic functions represented using function notation [e.g., evaluate f (21 ), given f(x) = 2x2 + 3x – 1], including functions
arising from real-world applications
Sample problem: The relationship between the selling price of a sleeping bag, s dollars, and the revenue at that selling price,
r(s) dollars, is represented by the function
r (s) = –10s2 + 1500s. Evaluate, interpret, and compare r (29.95), r (60.00), r (75.00), r (90.00), and r (130.00).
2.3 explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of lin- ear and quadratic functions, and describe the domain and range of a function appropriately (e.g.,fory=x2 +1,thedomainisthesetof real numbers, and the range is y ≥ 1)
2.4 explain any restrictions on the domain and the range of a quadratic function in contexts arising from real-world applications
Sample problem: A quadratic function repre-
sents the relationship between the height of a ball and the time elapsed since the ball was thrown. What physical factors will restrict the domain and range of the quadratic function?
2.5 determine, through investigation using technology, the roles of a, h, and k in quadratic functions of the form f(x) = a(x – h)2 + k, and describe these roles in terms of transforma- tions on the graph of f(x) = x2 (i.e., translations;
reflections in the x-axis; vertical stretches and compressions to and from the x-axis)
Sample problem: Investigate the graph f(x) = 3(x – h)2 + 5 for various values of h, using technology, and describe the effects of changing h in terms of a transformation.
2.6 sketch graphs of g(x) = a(x – h)2 + k by applying one or more transformations to the graph of f(x) = x2
Sample problem: Transform the graph of f(x)=x2 tosketchthegraphsofg(x)=x2 –4 andh(x)=–2(x+1)2.
2.7 express the equation of a quadratic function in the standard form f(x) = ax2 + bx + c, given the vertex form f(x) = a(x – h)2 + k, and verify, using graphing technology, that these forms are equivalent representations
Sample problem: Given the vertex form
f (x) = 3(x – 1)2 + 4, express the equation in standard form. Use technology to compare the graphs of these two forms of the equation.
2.8 express the equation of a quadratic function in the vertex form f(x) = a(x – h)2 + k, given the standard form f(x) = ax2 + bx + c, by completing the square (e.g., using algebra tiles or diagrams; algebraically), including
cases where ba is a simple rational number (e.g., 1 , 0.75), and verify, using graphing
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technology, that these forms are equivalent representations
2.9 sketch graphs of quadratic functions in the factored form f(x) = a(x – r )(x – s) by using the x-intercepts to determine the vertex
2.10 describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form f(x) = ax2 + bx + c, the vertex form f(x) = a(x – h)2 + k, and the factored form f(x) = a(x – r)(x – s) of a quadratic function
2.11 sketch the graph of a quadratic function whose equation is given in the standard formf(x)=ax2+bx+cbyusingasuitable 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)
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