Page 64 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

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1. simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;
2. identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;
3. demonstrate an understanding of compound interest and annuities, and solve related problems.
1. Connecting Graphs and Equations of Exponential Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
SPECIFIC EXPECTATIONS
and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases
or decreases throughout its domain) for
By the end of this course, students will:
exponential functions represented in a
variety of ways [e.g., tables of values, mapping 1.1 determine, through investigation using a
diagrams, graphs, equations of the form variety of tools (e.g., calculator, paper and
f(x) = ax (a > 0, a ≠ 1), function machines] pencil, graphing technology) and strategies
(e.g., patterning; finding values from a graph;
Sample problem: Graph f(x) = 2x, g(x) = 3x, interpreting the exponent laws), the value
and h(x) = 0.5x on the same set of axes.
of a power with a rational exponent (i.e., x mn ,
Make comparisons between the graphs, where x > 0 and m and n are integers)
and explain the relationship between the y-intercepts.
Sample problem: The exponent laws suggest
11
1.5
determine, through investigation (e.g., by that 42 x 42 = 41. What value would you
1
patterning with and without a calculator), assign to 42 ? What value would you assign
the exponent rules for multiplying and
13
dividing numeric expressions involving
to 27
? Explain your reasoning. Extend your
1 3 1 2
reasoning to make a generalization about the
exponents [e.g., ( ) x ( ) ], and the 122
meaning of xn, where x > 0 and n is a natural
exponent rule for simplifying numerical number.
expressions involving a power of a power
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B. EXPONENTIAL FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1.2 evaluate, with and without technology,
numerical expressions containing integer
and rational exponents and rational bases 1
[e.g., 2–3, (–6)3, 42 , 1.01120 ]
1.3 graph, with and without technology, an expo- nential relation, given its equation in the form y = ax (a > 0, a ≠ 1), define this relation as the function f(x) = ax, and explain why it is a function
1.4 determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals,
[e.g., 53 2], and use the rules to simplify ()
numerical expressions containing integer exponents [e.g., (23)(25) = 28]
1.6 distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations), within the same context when possible (e.g., simple interest and compound interest, population growth)