Page 71 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. MATHEMATICAL MODELS OVERALL EXPECTATIONS
By the end of this course, students will:
1. make connections between the numeric, graphical, and algebraic representations of quadratic relations, and use the connections to solve problems;
2. demonstrate an understanding of exponents, and make connections between the numeric, graphical, and algebraic representations of exponential relations;
3. describe and represent exponential relations, and solve problems involving exponential relations arising from real-world applications.
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.4 sketch graphs of quadratic relations repre- sented by the equation y = a(x – h)2 + k (e.g., using the vertex and at least one point on each side of the vertex; applying one or more transformations to the graph of y = x2)
1. Connecting Graphs and Equations of Quadratic Relations
1.1 construct tables of values and graph quadra-
1.5 expand and simplify quadratic expressions in tic relations arising from real-world applica-
one variable involving multiplying binomials tions (e.g., dropping a ball from a given
[e.g., (1 x + 1)(3x – 2)] or squaring a binomial height; varying the edge length of a cube 2
and observing the effect on the surface area
[e.g., 5(3x – 1)2], using a variety of tools (e.g., paper and pencil, algebra tiles, computer
of the cube)
algebra systems)
1.2 determine and interpret meaningful values
of the variables, given a graph of a quadratic
1.6 express the equation of a quadratic relation in relation arising from a real-world application
the standard form y = ax2 + bx + c, given the vertex form y = a(x – h)2 + k, and verify, using Sample problem: Under certain conditions,
graphing technology, that these forms are there is a quadratic relation between the
equivalent representations
profit of a manufacturing company and the
number of items it produces. Explain how you could interpret a graph of the relation
to determine the numbers of items produced for which the company makes a profit and to determine the maximum profit the company can make.
1.3 determine, through investigation using technology, the roles of a, h, and k in quadratic relations of the form y = a(x – h)2 + k, and describe these roles in terms of transforma- tions on the graph of y = x2 (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)
Sample problem: Investigate the graph y=3(x–h)2 +5forvariousvaluesofh, using technology, and describe the effects of changing h in terms of a transformation.
Sample problem: Given the vertex form
y = 3(x – 1)2 + 4, express the equation in standard form. Use technology to compare the graphs of these two forms of the equation.
1.7 factor trinomials of the form ax2 + bx + c, where a = 1 or where a is the common factor, by various methods
1.8 determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic relation
Sample problem: Investigate the relationship between the factored form of 3x2 + 15x + 12 and the x-intercepts of y = 3x2 + 15x + 12.
MATHEMATICAL MODELS
69
Foundations for College Mathematics
MBF3C