Page 95 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Sample problem: Investigate, using graphing technology, key features of the graphs of the family of rational functions of the form
f(x)= 8x forn=1,2,4,and8,andmake nx + 1
connections between the equations and the asymptotes.
2.3 sketch the graph of a simple rational function using its key features, given the algebraic rep- resentation of the function
By the end of this course, students will:
3.1 make connections, through investigation using technology (e.g., computer algebra systems), between the polynomial function f(x), the divisor x – a, the remainder from the division
f (x) , and f (a) to verify the remainder theorem x–a
and the factor theorem
Sample problem: Divide
f(x) = x4 + 4x3 – x2 – 16x – 14 by x – a for various integral values of a using a computer algebra system. Compare the remainder from each division with f(a).
3.2 factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factor- ing, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem)
Sample problem: Factor: x3 + 2x2 – x – 2; x4 – 6x3 + 4x2 + 6x – 5.
3.3 determine, through investigation using tech- nology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this con- nection [e.g., the real roots of the equation
x4 – 13x2 + 36 = 0 are the x-intercepts of the graph of f(x) = x4 – 13x2 + 36]
Sample problem: Describe the relationship between the x-intercepts of the graphs of linear and quadratic functions and the real roots of the corresponding equations. Investigate, using technology, whether this relationship exists for polynomial functions of higher degree.
3.4 solve polynomial equations in one variable, of degree no higher than four (e.g., 2x3–3x2+8x–12=0),byselectingand applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem), and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graph- ing technology to determine the x-intercepts of the graph of the corresponding polynomial function)
3.5 determine, through investigation using tech- nology (e.g., graphing calculator, computer algebra systems), the connection between
the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection
x–2 [e.g., the real root of the equation x – 3 = 0
is 2, which is the x-intercept of the function f(x)=x–2 ;theequation 1 =0hasno x–3 x–3 1
real roots, and the function f(x) = x – 3 does not intersect the x-axis]
3.6 solve simple rational equations in one variable algebraically, and verify solutions using tech- nology (e.g., using computer algebra systems to determine the roots; using graphing tech- nology to determine the x-intercepts of the graph of the corresponding rational function)
3.7 solve problems involving applications of polynomial and simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem, such as deter- mining the values of k for which the function f(x)=x3+6x2+kx–4givesthesameremain- der when divided by x – 1 and x + 2]
Sample problem: Use long division to express thegivenfunctionf(x)=x2+3x–5asthe x–1
sum of a polynomial function and a rational
function of the form A (where A is a x–1
constant), make a conjecture about the rela- tionship between the given function and the polynomial function for very large positive and negative x-values, and verify your con- jecture using graphing technology.
POLYNOMIAL AND RATIONAL FUNCTIONS
3. Solving Polynomial and Rational Equations
93
Advanced Functions
MHF4U