Page 47 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. CHARACTERISTICS OF FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1. demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;
2. determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications;
3. demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.
SPECIFIC EXPECTATIONS
1. Representing Functions
By the end of this course, students will:
1.1 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 repre- sentations (i.e., tables of values, mapping dia- grams, graphs, function machines, equations) and strategies (e.g., identifying a one-to-one or many-to-one mapping; using the vertical- line test)
Sample problem: Investigate, using numeric and graphical representations, whether the relation x = y 2 is a function, and justify your reasoning.
1.2 represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and
e v a l u a t e f u n c t i o n s [ e . g . , e v a l u a t e f ( 21 ) , g i v e n f(x)=2x2 +3x–1]
1.3 explain the meanings of the terms domain
and range, through investigation using numer- ic, graphical, and algebraic representations of the functions f(x) = x, f(x) = x2, f(x) = √x,
and f(x) = 1x; describe the domain and range of
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 quad- ratic function?
1.4 relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations)
1.5 determine the numeric or graphical represen- tation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections, through investigation using a variety of tools (e.g., graphing tech- nology, Mira, tracing paper), between the graph of a function and the graph of its inverse (e.g., the graph of the inverse is the reflection of the graph of the function in the
l i n e y = x )
Sample problem: Given a graph and a table of values representing population over time, produce a table of values for the inverse and graph the inverse on a new set of axes.
1.6 determine, through investigation, the relation- ship between the domain and range of a func- tion and the domain and range of the inverse relation, and determine whether or not the inverse relation is a function
Sample problem: Given the graph of f(x) = x2, graph the inverse relation. Compare the domain and range of the function with the domain
CHARACTERISTICS OF FUNCTIONS
a function appropriately (e.g., for y = x2 + 1,
the domain is the set of real numbers, and the Sample problem: A quadratic function repre- range is y ≥ 1); and explain any restrictions on sents the relationship between the height
the domain and range in contexts arising from of a ball and the time elapsed since the ball real-world applications
was thrown. What physical factors will
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Functions
MCR3U