Page 94 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
2. Connecting Graphs and Equations of Rational Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
one or more repeated factors, for example, f(x)=(x–2)(x–3), f(x)=(x–2)(x–2)(x–3), f(x) = (x – 2)(x – 2)(x – 2)(x – 3), and
f(x) = (x + 2)(x + 2)(x – 2)(x – 2)(x – 3),
by considering whether the factor is repeated an even or an odd number of times. Use your conclusions to sketch
f(x) = (x + 1)(x + 1)(x – 3)(x – 3), and verify using technology.
1.6 determine, through investigation using tech- nology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x – d )) + c, and describe these roles in terms of transforma- tions on the graphs of f(x) = x3 and f(x) = x4 (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the
x- and y-axes)
Sample problem: Investigate, using technol- ogy, the graph of f(x) = 2(x – d)3 +c for various values of d and c, and describe
the effects of changing d and c in terms of transformations.
1.7 determine an equation of a polynomial func- tion that satisfies a given set of conditions (e.g., degree of the polynomial, intercepts, points on the function), using methods appropriate to the situation (e.g., using the x-intercepts of the function; using a trial-and-error process with a graphing calculator or graphing soft- ware; using finite differences), and recognize that there may be more than one polynomial function that can satisfy a given set of condi- tions (e.g., an infinite number of polynomial functions satisfy the condition that they have three given x-intercepts)
Sample problem: Determine an equation for a fifth-degree polynomial function that inter- sects the x-axis at only 5, 1, and –5, and sketch the graph of the function.
1.8 determine the equation of the family of poly- nomial functions with a given set of zeros and of the member of the family that passes through another given point [e.g., a family of polynomial functions of degree 3 with
zeros 5, –3, and –2 is defined by the equation f(x) = k(x – 5)(x + 3)(x + 2), where k is a real number, k ≠ 0; the member of the family
that passes through (–1, 24) is
f(x) = –2(x – 5)(x + 3)(x + 2)]
Sample problem: Investigate, using graphing technology, and determine a polynomial function that can be used to model the func- tion f(x) = sin x over the interval 0 ≤ x ≤ 2π.
1.9 determine, through investigation, and compare the properties of even and odd polynomial functions [e.g., symmetry about the y-axis
or the origin; the power of each term; the number of x-intercepts; f (x) = f (– x) or
f (– x) = – f (x)], and determine whether a given polynomial function is even, odd, or neither
Sample problem: Investigate numerically, graphically, and algebraically, with and with- out technology, the conditions under which an even function has an even number of x-intercepts.
By the end of this course, students will:
2.1 determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make con- nections between the algebraic and graphical representations of these rational functions [e.g.,
make connections between f(x) = 1 x2 – 4
and its graph by using graphing technology and by reasoning that there are vertical asymptotes at x = 2 and x = –2 and a hori- zontal asymptote at y = 0 and that the func- tion maintains the same sign as f(x) = x2 – 4]
Sample problem: Investigate, with technology, the key features of the graphs of families of rational functions of the form
f(x)= 1 andf(x)= 1 , x + n x2 + n
where n is an integer, and make connections between the equations and key features of the graphs.
2.2 determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that have linear expres- sions in the numerator and denominator
[e.g.,f(x)= 2x , h(x)= x–2 ],and x – 3 3x + 4
make connections between the algebraic and graphical representations of these rational functions
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