Page 93 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

C. POLYNOMIAL AND RATIONAL FUNCTIONS
OVERALL EXPECTATIONS
By the end of this course, students will:
1. identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
2. identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
3. solve problems involving polynomial and simple rational equations graphically and algebraically;
4. demonstrate an understanding of solving polynomial and simple rational inequalities.
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 recognize a polynomial expression (i.e., a series of terms where each term is the product of a constant and a power of x with a non- negative integral exponent, such as
x3 – 5x2 + 2x – 1); recognize the equation of
a polynomial function, give reasons why it
is a function, and identify linear and quad- ratic functions as examples of polynomial functions
1.2 compare, through investigation using graph- ing technology, the numeric, graphical, and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., compare finite differences in tables of values; investigate the effect of the degree of a polynomial function on the shape of its graph and the maximum number of x-intercepts; investigate the effect of varying the sign of the leading coefficient on the end behaviour of the function for very large positive or nega- tive x-values)
Sample problem: Investigate the maximum number of x-intercepts for linear, quadratic, cubic, and quartic functions using graphing technology.
1.3 describe key features of the graphs of poly- nomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or nega- tive x-values)
Sample problem: Describe and compare the key features of the graphs of the functions f(x)=x, f(x)=x2, f(x)=x3, f(x)=x3 +x2, andf(x)=x3 +x.
1.4 distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, g(x) = 2x], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions
1.5 make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g.,
f(x) = 2(x – 3)(x + 2)(x – 1)] and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features (e.g., by determining intercepts and end beha- viour; by locating positive and negative regions using test values between and on either side of the x-intercepts)
Sample problem: Investigate, using graphing technology, the x-intercepts and the shapes of the graphs of polynomial functions with
POLYNOMIAL AND RATIONAL FUNCTIONS
1. Connecting Graphs and Equations of Polynomial Functions
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Advanced Functions
MHF4U