Mathematics

Grade 11, University/College PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics60equations using the quadratic formula. Howmany of the equations could you solve byfactoring?2. Connecting Graphs and Equationsof Quadratic FunctionsBy the end of this course, students will:2.1 explain the meaning of the term function, anddistinguish a function from a relation that isnot a function, through investigation of linearand quadratic relations using a variety ofrepresentations (i.e., tables of values, mappingdiagrams, graphs, function machines, equations)and strategies (e.g., using the verticallinetest)Sample problem: Investigate, using numericand graphical representations, whether the2relation x = y is a function, and justifyyour reasoning.2.2 substitute into and evaluate linear andquadratic functions represented using1function notation [e.g., evaluate f ( 2 ) , given2f(x) = 2x + 3x – 1], including functionsarising from real-world applicationsSample problem: The relationship betweenthe selling price of a sleeping bag, s dollars,and the revenue at that selling price,r(s) dollars, is represented by the function2r (s) = –10s + 1500s. Evaluate, interpret, andcompare 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 andrange, through investigation using numeric,graphical, and algebraic representations of linearand quadratic functions, and describe thedomain and range of a function appropriately2(e.g., for y = x + 1, the domain is the set ofreal numbers, and the range is y ≥ 1)2.4 explain any restrictions on the domain andthe range of a quadratic function in contextsarising from real-world applicationsSample problem: A quadratic function representsthe relationship between the height of aball and the time elapsed since the ball wasthrown. What physical factors will restrict thedomain and range of the quadratic function?2.5 determine, through investigation usingtechnology, the roles of a, h, and k in quadratic2functions of the form f(x) = a(x – h) + k, anddescribe these roles in terms of transformationson the graph of f(x) = x (i.e.,2translations;reflections in the x-axis; vertical stretches andcompressions to and from the x-axis)Sample problem: Investigate the graph2f(x) = 3(x – h) + 5 for various valuesof h, using technology, and describe theeffects of changing h in terms of atransformation.22.6 sketch graphs of g(x) = a(x – h) + k byapplying one or more transformations to2the graph of f(x) = xSample problem: Transform the graph of22f(x) = x to sketch the graphs of g(x) = x – 42and h(x) = – 2(x + 1) .2.7 express the equation of a quadratic function2in the standard form f(x) = ax + bx + c, given2the vertex form f(x) = a(x – h) + k, and verify,using graphing technology, that these formsare equivalent representationsSample problem: Given the vertex form2f(x) = 3(x – 1) + 4, express the equation instandard form. Use technology to comparethe graphs of these two forms of the equation.2.8 express the equation of a quadratic function2in the vertex form f(x) = a(x – h) + k, given2the standard form f(x) = ax + bx + c, bycompleting the square (e.g., using algebratiles or diagrams; algebraically), includingbcases where is a simple rational numbera(e.g.,1, 0.75), and verify, using graphing2technology, that these forms are equivalentrepresentations2.9 sketch graphs of quadratic functions in thefactored form f(x) = a(x – r )(x – s) by usingthe x-intercepts to determine the vertex2.10 describe the information (e.g., maximum,intercepts) that can be obtained by inspecting2the standard form f(x) = ax + bx + c, the2vertex form f(x) = a(x – h) + k, and thefactored form f(x) = a(x – r)(x – s) of aquadratic function2.11 sketch the graph of a quadratic functionwhose equation is given in the standard2form f(x) = ax + bx + c by using a suitablestrategy (e.g., completing the square andfinding the vertex; factoring, if possible, tolocate the x-intercepts), and identify the keyfeatures of the graph (e.g., the vertex, thex- and y-intercepts, the equation of the axisof symmetry, the intervals where the functionis positive or negative, the intervals wherethe function is increasing or decreasing)