Page 129 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

B. POLYNOMIAL FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1. recognize and evaluate polynomial functions, describe key features of their graphs, and solve problems using graphs of polynomial functions;
2. make connections between the numeric, graphical, and algebraic representations of polynomial functions;
3. solve polynomial equations by factoring, make connections between functions and formulas, and solve problems involving polynomial expressions arising from a variety of applications.
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 and 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 graphical and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., inves- tigate 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 negative 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 negative 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,andf(x)=x4.
1.4 distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, f(x) = 2x)], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions
1.5 substitute into and evaluate polynomial func- tions expressed in function notation, including functions arising from real-world applications
Sample problem: A box with no top is being made out of a 20-cm by 30-cm piece of cardboard by cutting equal squares of
side length x from the corners and folding up the sides. The volume of the box is
V = x(20 – 2x)(30 – 2x). Determine the volume if the side length of each square is 6 cm. Use the graph of the polynomial function V(x) to determine the size of square that should be cut from the corners if the required volume
of the box is 1000 cm3.
1.6 pose problems based on real-world applica- tions that can be modelled with polynomial functions, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
1.7 recognize, using graphs, the limitations of modelling a real-world relationship using a polynomial function, and identify and explain any restrictions on the domain and range (e.g., restrictions on the height and time for a
POLYNOMIAL FUNCTIONS
1. Investigating Graphs of Polynomial Functions
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Mathematics for College Technology
MCT4C